x + x x x(x - x) dx = - Vidyadip

MCQ

$\int {\frac{{\cos x + x\sin x}}{{x(x - \cos x)}}dx = } $

A
$\log |x(x - \cos x)| + c$
✓
$\log \left| {1 - \frac{{\cos x}}{x}} \right| + c$
C
$\log \left| {\frac{x}{{x - \cos x}}} \right| + c$
D
None of these

✓

Answer

Correct option: B.

$\log \left| {1 - \frac{{\cos x}}{x}} \right| + c$

b
$\int \frac{\cos x+x \sin x}{x^{2}\left(1-\frac{\cos x}{x}\right)} \cdot d x$

Put $1-\frac{\cos x}{x}=t$

$-\left[\frac{-x \sin x-\cos x}{x^{2}}\right] d x=d t$

$\frac{x \sin x+\cos x}{x^{2}} d x=d t$

$\int \frac{\mathrm{dt}}{\mathrm{t}}$

$\ln t+c$

$ = \ln \left| {1 - \frac{{\cos {\rm{x}}}}{{\rm{x}}}} \right| + {\rm{c}}$

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