Evaluate: $\int \cos ^{-1}(\sin x) d x$
Given, $\int \cos ^{-1}(\sin x) d x$
Let us consider, $\int \cos ^{-1} \mathrm{dx}$
We know that, $\int f(x) \cdot g(x) d x=f(x) \int g(x) d x-\int\left[f^{\prime}(x) \int g(x)\right] d x$
By comparison, $f(x)=\cos ^{-1} x ; g(x)=1$
$=\cos ^{-1} x x \int 1 d x-\int-\frac{1}{\sqrt{1-x^{2}}} \cdot x d x$
$=x \cos ^{-1} x-\frac{1}{2} \int \frac{1}{\sqrt{1-x^{2}}}(-2 x) d x$
$=x \cos ^{-1} x-\frac{1}{2} \int\left(1-x^{2}\right)^{-\frac{1}{2}}(-2 x) d x$
$=x \cos ^{-1} x-\frac{1}{2} \frac{\left(1-x^{2}\right)^{-\frac{1}{2}+1}}{\frac{1}{2}+1}+c\left(\right.$ since, $\left.\int\left[f(x)^{n} \cdot f^{I}(x)\right] d x=\frac{f(x)^{n+1}}{n+1}\right)$
$=x \cos ^{-1} x-\left(1-x^{2}\right)^{1 / 2}+c$
$=x \cos ^{-1} x-\sqrt{1-x^{2}}+c$
Therefore, $\int \cos ^{-1} x d x=x \cos ^{-1} x-\sqrt{1-x^{2}}+c$
Replace ' $x$ ' with ' $\sin x$ ' :-
$\int \cos ^{-1}(\sin x) d x=\sin x \cdot \cos ^{-1}(\sin x)-\sqrt{1-(\sin x)^{2}}+c$
$=\sin x \cdot \cos ^{-1} x(\sin x)-\cos x+c$
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