First slide

Methods of integration

Question

$Evaluate \int \sin (\log x) d x$

Moderate

A

$\frac{e^{x}}{2} [\sin (\log x) - \cos (\log x)] + c$
B

$\frac{x}{2} [\cos (\log x) - \sin (\log x)] + c$
C

$\frac{x}{2} [\sin (\log x) - \cos (\log x)] + c$
D

None of these

Solution

$Let I = \int \sin (\log x) d x Let \log x = t . Then x = e^{t} or d x = e^{t} d t . Thus. I = \int \sin t e^{t} d t = \frac{e^{t}}{2} (\sin t - \cos t) + c u \sin g t h e f o r m u l a \int e^{a x} \sin (b x + c) = \frac{e^{a x}}{a^{2} + b^{2}} [a \sin (b x + c) - b \cos (b x + c)] H e n c e, \int \sin (\log x) d x = \frac{x}{2} [\sin (\log x) - \cos (\log x)] + c$

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