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Integral Calculus> Integrate sin(tan^-1x)dx please simplify ...

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Integrate sin(tan^-1x)dx please simplify and then solve

Prathamesh Kulkarni , 7 Years ago

Grade 11

1 Answers

Anshuman Guru

Last Activity: 7 Years ago

$\int sin(tan^{-1}x)dx$

Let $tan^{-1}x=\theta$

=> tan θ = x

=> $\sec \theta =\sqrt{1+x^2}$

=> $\cos \theta =1/\sqrt{1+x^2}$

=> $\sin \theta =x/\sqrt{1+x^2}$

=> $\theta =\sin^{-1}(x/\sqrt{1+x^2})$

I= $\int \sin(\sin^{-1}(x/\sqrt{1+x^2}))dx$

= $\int (x/\sqrt{1+x^2}))dx$

let x²=t

=> 2xdx=dt

=> xdx=dt/2

I= $\frac{1}{2}\int \frac{dt}{\sqrt{1+t}}$

= $\sqrt{1+t} + c$

= $\sqrt{1+x^{2}} + c$

just convert tan-1 x to sin-1 x using angles or you can just do it by triangle method..... then as the derivative of denominator is present in numerator so either take x^2 or 1+x^2 as t and substitute accoringly

Hope this helps :-)

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