Mathematics

$$\displaystyle\int x^2\sin^{-1}x dx$$.


SOLUTION
$$\displaystyle \int x^2 \sin^{-1} x dx$$
Using byparts,we get
$$\displaystyle \cfrac{1}{3} x^3 \sin^{-1} x - \cfrac{1}{3} \int \cfrac{x^3}{\sqrt{1-x^2}}  dx$$
Putting $$x^2 = m $$
$$2x dx = dm$$
$$\displaystyle \cfrac{1}{3} x^3 \sin^{-1} x -\cfrac{1}{6} \int \cfrac{m}{\sqrt{1-m
}} dm$$
$$\displaystyle \cfrac{1}{3} x^3 \sin^{-1} x - \cfrac{1}{6} \int \cfrac{1-(1-m)}{\sqrt{1-m}}dm$$
$$\cfrac{1}{3} x^3 \sin^{-1} x - \cfrac{1}{3} \sqrt{1-x^2} + \cfrac{1}{9} \sqrt{(1-x^2)^3}$$
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Subjective Medium Published on 17th 09, 2020
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