find-arcsin-2x-1-4x-2-dx-

Question Number 45970 by maxmathsup by imad last updated on 19/Oct/18

f i n d \int \frac{a r c s i n (2 x)}{\sqrt{1 - 4 x^{2}}} d x

Commented by maxmathsup by imad last updated on 20/Oct/18

c h a n g e m e n t a r c s i n (2 x) = t g i v e 2 x = s i n t \Rightarrow 2 d x = c o s t d t \Rightarrow

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

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

Answered by last updated on 19/Oct/18

∵ {s i n}^{- 1} 2 x = t

\frac{1}{\sqrt{1 - 4 x^{2}}} d x = \frac{1}{2} d t

= \frac{1}{2} \int t d t

= \frac{1}{4} {[{s i n}^{- 1} 2 x]}^{2} + c

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