2009 Today's Calculation Of Integral

Part of Today's Calculation Of Integral

Subcontests

(125)

1

Today's calculation of Integral 521

t

be a positive number. Draw two tangent lines from the point (t, \minus{} 1) to the parabpla y \equal{} x^2. Denote

S(t)

the area bounded by the tangents line and the parabola. Find the minimum value of

\frac {S(t)}{\sqrt {t}}

1

Today's calculation of Integral 520

a,\ b,\ c

be postive constants. Evaluate \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx.

1

Today's calculation of Integral 519

Evaluate \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx.

1

Today's calculation of Integral 518

{ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx

1

Today's calculation of Integral 517

Consider points

P

which are inside the square with side length

a

such that the distance from

P

to the center of the square equals to the least distance from

P

to each side of the square.Find the area of the figure formed by the whole points

P

1

Today's calculation of Integral 516

Let f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0

f(x)

\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx

1

Today's calculation of Integral 515

Find the maximum and minimum values of \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx for

|a|\leq 1,\ |b|\leq 1

. Note that you are not allowed to solve in using partial differentiation here.

1

Today's calculation of Integral 514

Prove the following inequalities: (1) x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right) (2) \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)

1

Today's calculation of Integral 513

Find the constants

a,\ b,\ c

such that a function f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c satisfies the following equation for any real numbers

x

. 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).

1

Today's calculation of Integral 512

Evaluate \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).

1

Today's calculation of Integral 511

f(x),\ g(x)

are differential fuctions and their derivatives are continuous. Find

f(x),\ g(x)

such that f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt.

1

Today's calculation of Integral 510

(1) Evaluate \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx. (2) For f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt, find f''(x)\plus{}f(x).

1

Today's calculation of Integral 509

Evaluate \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx.

1

Today's calculation of Integral 508

Compare the size of the definite integrals?

\int_0^{\frac {\pi}{4}} x^{2008}\tan ^{2008}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2009}\tan ^{2009}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2010}\tan ^{2010}x\ dx

1

Today's calculation of Integral 507

Evaluate \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx

1

Today's calculation of Integral 506

a,\ b

be the real numbers such that

0\leq a\leq b\leq 1

. Find the minimum value of \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx.

1

Today's calculation of Integral 505

xyz

space with the origin

O

, given a cuboid

K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2

and the plane \alpha : z \equal{} 2. Draw the perpendicular

PH

P

to the plane. Find the volume of the solid formed by all points of

P

which are included in

K

\overline{OP}\leq \overline{PH}

1

Today's calculation of Integral 504

a,\ b

are positive constants. Determin the value of a positive number

m

such that the areas of four parts of the region bounded by two parabolas y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b and the line y\equal{}mx have equal area.

1

Today's calculation of Integral 503

Prove the following inequality. \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}

1

Today's calculation of Integral 502

0 < x < 1

, prove that (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}. (2) Find \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}.

1

Today's calculation of Integral 501

Find the volume of the uion

A\cup B\cup C

of the three subsets

A,\ B,\ C

xyz

space such that: A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\} B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\} C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}

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Today's calculation of Integral 500

a,\ b,\ c

be positive real numbers. Prove the following inequality. \int_1^e \frac {x^{a \plus{} b \plus{} c \minus{} 1}[2(a \plus{} b \plus{} c) \plus{} (c \plus{} 2a)x^{a \minus{} b} \plus{} (a \plus{} 2b)x^{b \minus{} c} \plus{} (b \plus{} 2c)x^{c \minus{} a} \plus{}(2a \plus{} b)x^{a \minus{} c} \plus{} (2b \plus{} c)x^{b \minus{} a} \plus{} (2c \plus{} a)x^{c \minus{} b}]}{(x^a \plus{} x^b)(x^b \plus{} x^c)(x^c \plus{} x^a)}\geq a \plus{} b \plus{} c. I have just posted 500 th post. Thank you for your cooperations, mathLinkers and AOPS users. I will keep posting afterwards. Japanese Communities Modeartor kunny

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Today's calculation of Integral 499

Evaluate \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.

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Today's calculation of Integral 498

f(x)

be a continuous function defined in the interval

0\leq x\leq 1.

Prove that \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.

1

Today's calculation of Integral 497

Consider a parameterized curve C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right). (1) Find the length

L

C

. (2) Find the area

S

of the region bounded by

C

x

y

axis. You may not use the formula

\boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }

1

Today's calculation of Integral 496

Evaluate \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0). You may not use \tan ^{ \minus{} 1} x or Complex Integral here.

1

Today's calculation of Integral 495

Evaluate the following definite integrals. (1) \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx (2) \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx (3) \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}

1

Today's calculation of Integral 494

Suppose the curve C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0) has a common tangent line at the point

P

with the hyperbola xy \equal{} 1 in the first quadrant. (1) Find the value of

a

and the coordinate of the point

P

. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment

OP

C

OP

1

Today's calculation of Integral 493

In the x \minus{} y plane, let

l

be the tangent line at the point

A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)

on the ellipse \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a). Let denote

S

be the area of the figure bounded by

l,

x

axis and the ellipse. (1) Find the equation of

l

S

a,\ b

. (3) Find the maximum value of

S

with the constraint a^2 \plus{} 3b^2 \equal{} 4.

1

Today's calculation of Integral 492

Find the volume formed by the revolution of the region satisfying 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q) in the coordinate plane about the

y

-axis. You are not allowed to use the formula: V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)} here.

1

Today's calculation of Integral 491

Let f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x. (1) Evaluate \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx. (2) Find the area of the region bounded by two curves y\equal{}f(x) and y\equal{}g(x)\ (0\leq x\leq \pi).

1

Today's calculation of Integral 490

For a positive real number

a > 1

, prove the following inequality. \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}

1

Today's calculation of Integral 489

Find the following limit. \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx.

1

Today's calculation of Integral 488

0\leq x <\frac{\pi}{2}

, prove the following inequality. x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}

1

Today's calculation of Integral 487

Suppose two functions f(x)\equal{}x^4\minus{}x,\ g(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d satisfy f(1)\equal{}g(1),\ f(\minus{}1)\equal{}g(\minus{}1). Find the values of

a,\ b,\ c,\ d

such that \int_{\minus{}1}^1 (f(x)\minus{}g(x))^2dx is minimal.

1

Today's calculation of Integral 486

H

be the piont of midpoint of the cord

PQ

that is on the circle centered the origin

O

1.

Suppose the length of the cord

PQ

2\sin \frac {t}{2}

t\ (0\leq t\leq \pi)

that is formed by half-ray

OH

and the positive direction of the

x

axis. Answer the following questions. (1) Express the coordiante of

H

t

t

moves in the range of

0\leq t\leq \pi

, find the minimum value of

x

H

t

moves in the range of

0\leq t\leq \frac {\pi}{2}

, find the area

S

of the region bounded by the curve drawn by the point

H

x

y

1

Today's calculation of Integral 485

x

y

plane, for the origin

O

, given an isosceles triangle

OAB

with AO \equal{} AB such that

A

is on the first quadrant and

B

x

axis. Denote the area by

s

. Find the area of the common part of the traingle and the region expressed by the inequality

xy\leq 1

to give the area as the function of

s

1

Today's calculation of Integral 484

C: y=\ln x

. For each positive integer

n

A_n

the area of the part enclosed by the line passing through two points

(n,\ \ln n),\ (n+1,\ \ln (n+1))

B_n

that of the part enclosed by the tangent line at the point

(n,\ \ln n)

C

x=n+1

g(x)=\ln (x+1)-\ln x

A_n,\ B_n

n,\ g(n)

respectively.
(2) Find

\lim_{n\to\infty} n\{1-ng(n)\}

1

Today's calculation of Integral 483

n\geq 2

be natural number. Answer the following questions. (1) Evaluate the definite integral

\int_1^n x\ln x\ dx.

(2) Prove the following inequality. \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n. (3) Find

\lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.

1

Today's calculation of Integral 482

n

be natural number. Find the limit value of

{ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).

1

Today's calculation of Integral 481

For real numbers

a,\ b

|a|\neq |b|

, let I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2). Prove that : \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}

1

Today's calculation of Integral 480

a,\ b

be positive real numbers. Prove that \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx \leq \frac {\pi}3 (a^2 \plus{} b^2). Edited by moderator.

1

Today's calculation of Integral 479

a,\ b

be real constants. Find the minimum value of the definite integral: I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.

1

Today's calculation of Integral 478

Evaluate \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.

1

Today's calculation of Integral 477

Suppose that P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3. Find all possible values for which \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3) can be valued.

1

Today's calculation of Integral 476

Suppose a parabola with the axis as the

y

axis, concave up and touches the graph y\equal{}1\minus{}|x|. Find the equation of the parabola such that the area of the region surrounded by the parabola and the

x

axis is maximal.

1

Today's calculation of Integral 475

For a positive constant number

t

D

the region surrounded by the curve y \equal{} e^{x}, the line x \equal{} t, the

x

y

V_x,\ V_y

be the volumes of the solid obtained by rotating

D

x

y

axis respectively. Compare the size of

V_x,\ V_y.

1

Today's calculation of Integral 474

Calculate the following indefinite integrals. (1) \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx (2)

\int \sin 2x\cos 2x\cos 4x\ dx

\int xe^{x}dx

\int 5^{x}dx

1

Today's calculation of Integral 473

For nonzero real numbers

r,\ l

and the positive constant number

c

, consider the curve on the

xy

plane : y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r) \\ r^2 & (r\leq x\leq l \plus{} r) \\ (x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r) \end{array} \right. Denote

V

the volume of the solid by revolvering the curve about the

x

r,\ l

vary in such a way that r^2 \plus{} l \equal{} c. Find the values of

r,\ l

which gives the maxmimum volume.

1

Today's calculation of Integral 472

Given a line segment

PQ

moving on the parabola y \equal{} x^2 with end points on the parabola. The area of the figure surrounded by

PQ

and the parabola is always equal to

\frac {4}{3}

. Find the equation of the locus of the mid point

M

PQ

1

Today's calculation of Integral 471

Evaluate \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx.

1

Today's calculation of Integral 470

Determin integers

m,\ n\ (m>n>0)

for which the area of the region bounded by the curve y\equal{}x^2\minus{}x and the lines y\equal{}mx,\ y\equal{}nx is

\frac{37}{6}

1

Today's calculation of Integral 469

Evaluate \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt.

1

Today's calculation of Integral 468

Evaluate \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx.

1

Today's calculation of Integral 467

Let the curve C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0). (1) Find the maximum value of

y

. (2) Find the area of the figure bounded by the curve

C

x

axis. (3) Find the volume of the solid generated by rotation of the figure about the

y

1

Today's calculation of Integral 466

For n \equal{} 1,\ 2,\ 3,\ \cdots, let

(p_n,\ q_n)\ (p_n > 0,\ q_n > 0)

be the point of intersection of y \equal{} \ln (nx) and \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1. (1) Show that 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2} to find

\lim_{n\to\infty} q_n

\lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx

1

Today's calculation of Integral 465

Compute \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots) , then use this result, prove that \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e.

1

Today's calculation of Integral 464

Evaluate \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx.

1

Today's calculation of Integral 463

\int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx

1

Today's calculation of Integral 462

Evaluate \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx.

1

Today's calculation of Integral 461

Let I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots). (1) Find

I_1,\ I_2

\lim_{n\to\infty} I_n

1

Today's calculation of Integral 460

\int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx.

1

Today's calculation of Integral 459

Find \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots).

1

Today's calculation of Integral 458

S(t)

be the area of the traingle

OAB

with O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t). Evaluate

\int_1^ e S(t)^2\ln t\ dt

1

Today's calculation of Integral 457

Evaluate \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta

1

Today's calculation of Integral 456

Find \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}

1

Today's calculation of Integral 455

(1) Evaluate \int_1^{3\sqrt{3}} \left(\frac{1}{\sqrt[3]{x^2}}\minus{}\frac{1}{1\plus{}\sqrt[3]{x^2}}\right)\ dx. (2) Find the positive real numbers

a,\ b

t>1,

\lim_{t\rightarrow \infty} \left(\int_1^t \frac{1}{1\plus{}\sqrt[3]{x^2}}\ dx\minus{}at^b\right) converges.

1

Today's calculation of Integral 454

a

be positive constant number. Evaluate \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.

1

Today's calculation of Integral 453

Find the minimum value of \int_0^{\frac{\pi}{2}} |x\sin t\minus{}\cos t|\ dt\ (x>0).

1

Today's calculation of Integral 452

a,\ b

are postive constant numbers. (1) Differentiate \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0). (2) For a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}, evaluate \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.

1

Today's calculation of Integral 451

Find \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).

1

Today's calculation of Integral 450

a,\ b

be postive real numbers. Find \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.

1

Today's calculation of Integral 449

Evaluate \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.

1

Today's calculation of Integral 448

Evaluate \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.

1

Today's calculation of Integral 447

Evaluate \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.

1

Today's calculation of Integral 445

Evaluate \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.

1

Today's calculation of Integral 444

Evaluate \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.

1

Today's calculation of Integral 443

Evaluate \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.

1

Today's calculation of Integral 442

Evaluate \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta

1

Today's calculation of Integral 441

Evaluate \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.

1

Today's calculation of Integral 440

a>1

, find \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.

1

Today's calculation of Integral 439

Find the volume of the solid defined by the inequality x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2. Note that you may not directively use double integral here for Japanese high school students who don't study it.

1

Today's calculation of Integral 438

Evaluate \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.

1

Today's calculation of Integral 437

Evaluate \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.

1

Today's calculation of Integral 436

Find the minimum area bounded by the graphs of y\equal{}x^2 and y\equal{}kx(x^2\minus{}k)\ (k>0).

1

Today's calculation of Integral 435

Evaluate \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx.

1

Today's calculation of Integral 434

Evaluate \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx.

1

Today's calculation of integral 433

Evaluate \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx.

1

Today's calculation of Integral 432

Define the function f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx. Find the minimum value of

f(t)

1\leq t\leq e

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Today's calculation of Integral 431

Consider the function f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx in the interval of

0\leq \theta \leq \frac {\pi}{2}

. (1) Find the maximum and minimum values of

f(\theta)

\int_0^{\frac {\pi}{2}} f(\theta)\ d\theta

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Today's calculation of Integral 430

For a natural number

n

, let a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx. Answer the following questions. (1) Find

a_1

. (2) Express a_{n\plus{}1} in terms of

a_n

\lim_{n\to\infty} a_n

. (4) Find \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}.

1

Today's calculation of Integral 429

Find the length of the curve expressed by the polar equation: r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi).

1

Today's calculation of Integral 428

f(x)

be a polynomial and

C

be a real number. Find the

f(x)

C

such that \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C.

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Today's calculation of Integral 427

a

be a positive real number, in Euclidean space, consider the two disks: D_1\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}a\}, D_2\equal{}\{(x,\ y,\ z)| x^2\plus{}y^2\leq 1,\ z\equal{}\minus{}a\}. Let

D_1

D_2

D_1

y

180^\circ

. Note that the rotational direction is supposed to be the direction such that we would lean the postive part of the

z

axis to into the direction of the postive part of

x

axis. Let denote

E

the part in which

D_1

passes while the rotation, let denote

V(a)

E

W(a)

be the volume of common part of

E

\{(x,\ y,\ z)|x\geq 0\}

W(a)

\lim_{a\rightarrow \infty} V(a)

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Today's calculation of Integral 426

Consider the polynomial f(x) \equal{} ax^2 \plus{} bx \plus{} c, with degree less than or equal to 2. When

f

varies with subject to the constrain f(0) \equal{} 0,\ f(2) \equal{} 2, find the minimum value of S\equal{}\int_0^2 |f'(x)|\ dx.

1

Today's calculation of Integral 425

The coordinate of

P

t

, moving on a plane, is expressed by

x = f(t) = \cos 2t + t\sin 2t,\ y = g(t) = \sin 2t - t\cos 2t

. (1) Find the acceleration vector

\overrightarrow{\alpha}

P

t

L

denote the line passing through the point

P

t%Error. "neqo" is a bad command.

, which is parallel to the acceleration vector

\overrightarrow{\alpha}

at the time. Prove that

L

always touches to the unit circle with center the origin, then find the point of tangency

Q

. (3) Prove that

f(t)

decreases in the interval

0\leq t \leqq \frac {\pi}{2}

t

varies in the range

\frac {\pi}{4}\leq t\leq \frac {\pi}{2}

, find the area

S

of the figure formed by moving the line segment

PQ

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Today's calculation of Integral 424

n

be positive integer. For n \equal{} 1,\ 2,\ 3,\ \cdots n, let denote

S_k

\triangle{AOB_k}

such that \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k. Find the limit \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k.

1

Today's calculation of Integral 423

Let f(x)\equal{}x^2\plus{}3 and y\equal{}g(x) be the equation of the line with the slope

a

, which pass through the point

(0,\ f(0))

. Find the maximum and minimum values of I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx.

1

Today's calculation of Integral 422

There are 10 cards, labeled from 1 to 10. Three cards denoted by

a,\ b,\ c\ (a > b > c)

are drawn from the cards at the same time. Find the probability such that \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0.

1

Today's calculation of Integral 421

Let f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x for real number

p > 0

. Answer the following questions. (1) Find the value of x \equal{} s_p for which

f(x)

is minimal and draw the graph of y \equal{} f(x). (2) Let g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx. Find the value of t \equal{} t_p for which

g(t)

is minimal. (3) Use the fact 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1) to find the limit \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p).

1

Today's calculation of Integral 420

K

be the figure bounded by the curve y\equal{}e^x and 3 lines x\equal{}0,\ x\equal{}1,\ y\equal{}0 in the

xy

plane. (1) Find the volume of the solid formed by revolving

K

x

axis. (2) Find the volume of the solid formed by revolving

K

y

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Today's calculation of Integral 419

xy

plane, the line

l

touches to 2 parabolas y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax, where

a

is positive constant. (1) Find the equation of

l

. (2) Find the area

S

bounded by the parabolas and the tangent line

l

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Today's calculation of Integral 418

(1) 2009 Kansai University entrance exam Calculate \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx. (2) 2009 Rikkyo University entrance exam/Science Evaluate \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx.

1

Today's calculation of Integral 417

f(x) ,\ g(x)

satify that f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt. Let

l_1,\ l_2

be the tangent lines of the curve y \equal{} f(x), which pass through the point

(a,\ g(a))

on the curve y \equal{} g(x). Find the minimum area of the figure bounded by the tangent tlines

l_1,\ l_2

and the curve y \equal{} f(x) .

1

Today's calculation of Integral 416

Answer the following questions. (1)

0 < x\leq 2\pi

|\sin x| < x

. (2) Let f_1(x) \equal{} \sin x\ , a be the constant such that

0 < a\leq 2\pi

. Define f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots). Find

f_2(x)

f_n(x)

n

. (4) For a given

x

, find \sum_{n \equal{} 1}^{\infty} f_n(x).

1

Today's calculation of Integral 415

For a function f(x) \equal{} 6x(1 \minus{} x), suppose that positive constant

c

and a linear function g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0) satisfy the following 3 conditions: c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0. Answer the following questions. (1) Find the constants

a,\ b,\ c

. (2) For natural number

n

, let I_n \equal{} \int_0^1 x^ne^x\ dx. Express I_{n \plus{} 1} in terms of

I_n

. Then evaluate

I_1,\ I_2,\ I_3

. (3) Evaluate the definite integrals

\int_0^1 e^xf(x)\ dx

\int_0^1 e^xf(x)g(x)\ dx

. (4) For real numbers

s,\ t

, define J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx. Find the constants

A,\ B,\ C,\ D,\ E

by setting J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F. (You don't need to find the constant

F

). (5) Find the values of

s,\ t

J

1

Today's calculation of Integral 414

Evaluate \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx.

1

Today's calculation of Integral 413

Find the maximum and minimum value of F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt for

0\leq x\leq \pi

1

Today's calculation of Integral 412

Let the definite integral I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots ). (1) Find I_0,\ I_{\minus{}1},\ I_2. (2) Find

I_1

. (3) Express I_{n\plus{}2} in terms of

I_n

. (4) Find I_{\minus{}3},\ I_{\minus{}2},\ I_3. (5) Evaluate the definite integrals \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx in using the avobe results. You are not allowed to use the formula of integral for \sqrt{x^2\plus{}1} directively here.

1

Today's calculation of Integral 411

Find the area bounded by y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7 and the

x

1

Today's calculation of Integral 410

Evaluate \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta.

1

Today's calculation of Integral 409

Evaluate \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx.

1

Today's calculation of Integral 408

Evaluate \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx.

1

Today's calculation of Integral 407

Evaluate \int_0^1 (x \plus{} 3)\sqrt {xe^x}\ dx.

1

Today's calculation of Integral 406

Find \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx.

1

Today's calculation of Integral 405

\displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|

1

Today's calculation of Integral 404

Evaluate \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots).

1

Today's calculation of Integral 403

Evaluate \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx.

1

Today's calculation of Integral 402

Consider a right circular cylinder with radius

r

of the base, hight

h

. Find the volume of the solid by revolving the cylinder about a diameter of the base.

1

Today's calculation of Integral 401

For real number

a

|a|>1

, evaluate \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}.

1

Today's calculation of Integral 400

(1) A function is defined f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2}) for

x\geq 0

f'(x)

. (2) Find the arc length of the part

0\leq \theta \leq \pi

for the curve defined by the polar equation: r \equal{} \theta\ (\theta \geq 0). Remark: You may not directly use the integral formula of \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2} here.

1

Today's calculation of Integral 399

Evaluate \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx.

1

Today's calculation of Integral 398

xyz

space, find the volume of the solid expressed by the sytem of inequality:

0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1

x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0

1

Today's calculation of Integral 397

xy

plane, find the minimum volume of the solid by rotating the region boubded by the parabola y \equal{} x^2 \plus{} ax \plus{} b passing through the point (1,\ \minus{} 1) and the

x

x

1

Today's calculation of Integral 396

Evaluate \int_0^{2008} \left(3x^2 \minus{} 8028x \plus{} 2007^2 \plus{} \frac {1}{2008}\right)\ dx.