$\int e^{\cos ^{-1} x}\left[\frac{x-\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}\right] d x=$

Question

Aicharya · Accepted Answer

Put $\cos ^{-1} x=t \Rightarrow \frac{-1}{\sqrt{1-x^{2}}} d x=d t \Rightarrow \frac{1}{\sqrt{1-x^{2}}} d x=-d t$ and $x=\cos t$ $\begin{aligned} I &=-\int e^{t}[\cos t-\sin t] d t=-e^{t} \cos t+c \ &=-e^{\cos ^{-1} x} \cos \left(\cos ^{-1} xight)+c=-x e^{\cos ^{-1} x}+c \end{aligned}$

$\int e^{\cos ^{-1} x}\left[\frac{x-\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}\right] d x=$

Solution