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  4. Value of the definite integral ∫ limits-1 / 21 / 2( sin -1(3 x-4 x3)- cos -1(4 x3-3 x)) d x

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Q. Value of the definite integral $\int\limits_{-1 / 2}^{1 / 2}\left(\sin ^{-1}\left(3 x-4 x^3\right)-\cos ^{-1}\left(4 x^3-3 x\right)\right) d x$

Integrals

Solution:

Note that in $\left(-\frac{1}{2}, \frac{1}{2}\right), \sin ^{-1}\left(3 x-4 x^3\right)=3 \sin ^{-1} x$ and $\cos ^{-1}\left(4 x^3-3 x\right)=2 \pi-3 \cos ^{-1} x$ hence
$f(x)=3 \sin ^{-1} x-2 \pi+3 \cos ^{-1} x=-\frac{\pi}{2} $
$\therefore I=-\frac{\pi}{2} \int\limits_{-1 / 2}^{1 / 2} d x=-\frac{\pi}{2}$