Question:
Evaluate $\int \cos ^{-1}\left(1-2 \mathrm{x}^{2}\right) \mathrm{dx}$
Solution:
Put $x=\sin t$
$; d x=\cos t d t$
$\int \cos ^{-1}\left(1-2 x^{2}\right) d x=\int \cos ^{-1}\left(1-2 \sin ^{2} t\right) \cos t d t=\int \cos ^{-1}\left(1-\sin ^{2} t-\sin ^{2} t\right) \cos t d t$
$\int \cos ^{-1}\left(\cos ^{2} t-\sin ^{2} t\right) \cos t d t=\int \cos ^{-1}(\cos 2 t) \cos t d t$
$2 \int t \cos t d t=2[t \sin t+\cos t]+c$
Ans $=2 \mathrm{x} \sin ^{-1} x+2 \sqrt{1-x^{2}}+c$
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