x-sin-1-x-dx-

Question Number 84666 by jagoll last updated on 14/Mar/20

\int x \sin^{- 1} (x) d x

Commented by mathmax by abdo last updated on 15/Mar/20

A = \int x a r c s i n x d x b y p a r t s u^{^{'}} = x a n d v = a r c s i n x \Rightarrow

A = \frac{x^{2}}{2} a r c s i n x - \int \frac{x^{2}}{2} \frac{d x}{\sqrt{1 - x^{2}}} w e h a v e

\int \frac{x^{2} d x}{\sqrt{1 - x^{2}}} = \int \frac{x^{2} - 1 + 1}{\sqrt{1 - x^{2}}} d x = \int \sqrt{1 - x^{2}} + \int \frac{d x}{\sqrt{1 - x^{2}}}

= - \int \sqrt{1 - x^{2}} d x + a r c s i n x + C b u t

\int \sqrt{1 - x^{2}} d x =_{x = s i n t} \int c o s t c o s t d t = \int \frac{1 + c o s (2 t)}{2} d t

= \frac{1}{2} t + \frac{1}{4} s i n (2 t) = \frac{1}{2} t + \frac{1}{2} s i n t c o s t = \frac{1}{2} a r c s i n x + \frac{1}{2} x \sqrt{1 - x^{2}}

\Rightarrow A = \frac{1}{2} x^{2} a r c s i n x - \frac{1}{2} (a r c s i n x - \frac{1}{2} a r c s i n x - \frac{1}{2} x \sqrt{1 - x^{2}}) + C

= \frac{x^{2}}{2} a r c s i n x - \frac{1}{4} a r c s i n x + \frac{1}{4} x \sqrt{1 - x^{2}} + C

= (\frac{x^{2}}{2} - \frac{1}{4}) a r c s i n x - \frac{x}{4} \sqrt{1 - x^{2}} + C

Commented by mathmax by abdo last updated on 15/Mar/20

A = (\frac{x^{2}}{2} - \frac{1}{4}) a r c s i n x + \frac{x}{4} \sqrt{1 - x^{2}} + C

Answered by MJS last updated on 15/Mar/20

\int x \arcsin x d x =

by parts

u = \arcsin x \to u^{'} = \frac{1}{\sqrt{1 - x^{2}}}

v^{'} = x \to v = \frac{1}{2} x^{2}

= \frac{1}{2} x^{2} \arcsin x - \frac{1}{2} \int \frac{x^{2}}{\sqrt{1 - x^{2}}} d x =

\int \frac{x^{2}}{\sqrt{1 - x^{2}}} d x =

[t = \arcsin x \to d x = \cos t d t]

= \int \sin^{2} t d t = \frac{1}{4} (2 t - \sin 2 t) =

= \frac{1}{2} (\arcsin x - x \sqrt{1 - x^{2}})

= \frac{1}{4} ((2 x^{2} - 1) \arcsin x + x \sqrt{1 - x^{2}}) + C

Commented by jagoll last updated on 15/Mar/20

thank you sir

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