What is the integration of x tan inverse x dx ?

Solution :

Let I = \(\int\) x\(tan^{-1}x\) dx

By using Integration by parts rule,

Taking tan inverse x as first function and x as second function. Then,

I = (\(tan^{-1}x\)) \(\int\) x dx โ€“ \(\int\){\({d\over dx}\)(\(tan^{-1}x\) \(\int\) x dx} dx

I = (\(tan^{-1}x\))\(x^2\over 2\) โ€“ \(\int\)\({1\over 1 + x^2}\) \(\times\) \(x^2\over 2\) dx

\(\implies\) I = \(x^2\over 2\)\(tan^{-1}x\) โ€“ \(1\over 2\) \(\int\) \(x^2 + 1 โ€“ 1\over x^2 + 1\) dx

\(\implies\) I = \(x^2\over 2\)\(tan^{-1}x\) โ€“ \(1\over 2\) \(\int\) 1 โ€“ \({1\over x^2 + 1}\) dx

\(\implies\) I = \(x^2\over 2\)\(tan^{-1}x\) โ€“ \(1\over 2\) (\(x โ€“ tan^{-1}x\)) + C


Similar Questions

What is the integration of tan inverse root x ?

Prove that \(\int_{0}^{\pi/2}\) log(sinx)dx = \(\int_{0}^{\pi/2}\) log(cosx)dx = -\(\pi\over 2\)log 2.

Evaluate : \(\int\) \(cos^4xdx\over {sin^3x{(sin^5x + cos^5x)^{3\over 5}}}\)

Evaluate : \(\int\) \(dx\over {3sinx + 4cosx}\)

Leave a Comment

Your email address will not be published. Required fields are marked *

Ezoicreport this ad