MathDB

97

Part of 2006 Today's Calculation Of Integral

Problems(1)

Today's calculation of integral 97

Source: Keio University/Science and Technology entrance exam 2006 Feb.

2/14/2006
Answer the following questions. (1) Evaluate ∫eeeln⁡(ln⁡x)xln⁡xdx.\int_e^{e^e} \frac{\ln (\ln x)}{x\ln x} dx. (2) Let α,β\alpha,\beta be real numbers.Find the values of α,β\alpha,\beta for which the following equality holds for any real numbers p,q.p,q. ∫−π2π2(pcos⁡x+qsin⁡x)(x2+αx+β)dx=0. \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(p\cos x+q\sin x)(x^2+\alpha x+\beta) dx=0.
calculusintegrationlogarithmstrigonometrycalculus computations