Question:
Evaluate the following integrals:
$\int \frac{1+\cos x}{(x+\sin x)^{3}} d x$
Solution:
Assume $x+\sin x=t$
$d(x+\sin x)=d t$
$(1+\cos x) d x=d t$
Substituting $\mathrm{t}$ and $\mathrm{dt}$ in given equation
$\Rightarrow \int \frac{\mathrm{dt}}{\mathrm{t}^{2}}$
$\Rightarrow \int \mathrm{t}^{-3} \mathrm{dt}$
$\Rightarrow \frac{t^{-2}}{-2}+c$
$\Rightarrow \frac{-1}{2 t^{2}}+c$
But $t=x+\sin x$
$\Rightarrow \frac{-1}{2(x+\sin x)^{2}}+c$
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