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Today's calculation of Integral 835

Source: 2012 Miyazaki University entrance exam/Engineering

July 8, 2012
calculusintegrationlogarithmstrigonometrycalculus computations

Problem Statement

Evaluate the following definite integrals.
(a) ∫12x−1x2−2x+2 dx\int_1^2 \frac{x-1}{x^2-2x+2}\ dx
(b) ∫01e4xe2x+2 dx\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx
(c) ∫1exln⁡x dx\int_1^e x\ln \sqrt{x}\ dx
(d) ∫0π3(cos⁡2xsin⁡3x−14sin⁡5x) dx\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx
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