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

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

Grade:11

1 Answers

Anshuman Guru
40 Points
4 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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