Mathematics

$$\int x  \sin ^ { - 1 } x \cdot d x$$


SOLUTION

$$\\\int x sin^{-1}x dx\\=sin^{-1}x\times(\frac{x^2}{2})-\int(\frac{1}{\sqrt{1-x^2}})\times(\frac{x^2}{2}) dx\\=(\frac{x^2}{2})sin^{-1}x-(\frac{1}{2})\int(\frac{x^2}{\sqrt{1-x^2}})dx\\=(\frac{x^2}{2})sin^{-1}x+(\frac{1}{2})\int\left[(\frac{(1-x^2)-1}{\sqrt{1-x^2}})\right]dx\\=(\frac{x^2}{2})sin^{-1}x+(\frac{1}{2})\int\sqrt{1-x^2}dx-(\frac{1}{2})\int(\frac{1}{\sqrt{1-x^2}})dx\\=(\frac{x^2}{2})sin^{-1}x+(\frac{1}{2})\left[(\frac{1}{2})\left(x\sqrt{1-x^2}+sin^{-1}x\right)\right]-(\frac{1}{2})sin^{-1}x+C\\=(\frac{2x^2-1}{4})sin^{-1}x+(\frac{x}{4})\sqrt{1-x^2}+C$$

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Subjective Medium Published on 17th 09, 2020
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