MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Today's Calculation Of Integral
2006 Today's Calculation Of Integral
163
163
Part of
2006 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 163
Source: created by kunny
10/21/2006
Let
I
n
=
∫
0
π
4
tan
n
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
.
I_{n}=\int_{0}^{\frac{\pi}{4}}\tan^{n}x\ dx\ (n=0,\ 1,\ 2,\ \cdots).
I
n
=
∫
0
4
π
tan
n
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
.
Find
∑
n
=
0
∞
{
I
n
+
2
2
+
(
I
n
+
1
+
I
n
+
3
)
I
n
+
2
+
I
n
+
1
I
n
+
3
}
.
\sum_{n=0}^{\infty}\{{I_{n+2}}^{2}+(I_{n+1}+I_{n+3})I_{n+2}+I_{n+1}I_{n+3}\}.
∑
n
=
0
∞
{
I
n
+
2
2
+
(
I
n
+
1
+
I
n
+
3
)
I
n
+
2
+
I
n
+
1
I
n
+
3
}
.
calculus
integration
trigonometry
calculus computations