First slide

Introduction to integration

Question

$\int \sin (\log x) d x = f (x) {\sin (\log x) - \cos (\log x)} + c$ , then $f (x) =$

Moderate

A

$2 x$
B

$x$
C

$x / 2$
D

None

Solution

$I = \int \sin (\log x) d x$
Put $\log x = t \Rightarrow x = e^{t}, d x = e^{t} d t$

$∴ I = \int e^{t} \sin t d t = \sin t e^{t} - \int (\cos t e^{t}) d t$

$= \sin t e^{t} - {\cos t e^{t} - \int (- \sin t) e^{t} d t}$

$= \sin t e^{t} - \cos t e^{t} - \int \sin t e^{t} d t$

$\Rightarrow 2 I = (\sin t - \cos t) e^{t} + c$

$\Rightarrow I = \frac{1}{2} x$ $(\sin (\log x) - \cos (\log x)) + c$ .

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