Correct Answer - Option 4 :
\(\rm {1\over2}\left[(x^2+1)\tan^{-1}x - x\right]+c\)
Concept:
Integration by parts: Integration by parts is a method to find integrals of products.
The formula for integrating by parts is given by:
⇒ \(\rm ∫ u vdx=u ∫ vdx- ∫ \left({du\over dx}\times ∫ vdx\right)dx \) + C
where u is the function u(x) and v is the function v(x)
ILATE rule is Usually, the preferred order for this rule and is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.
Calculation:
I = \(\rm \int x\tan^{-1}x dx\)
I = \(\rm \tan^{-1}x\int xdx - \int \left({1\over1+x^2}\int xdx\right)dx+c\)
I = \(\rm {x^2\tan^{-1}x\over2} - {1\over2}\int \left({x^2\over1+x^2}\right)dx+c\)
I = \(\rm {x^2\tan^{-1}x\over2} - {1\over2}\int \left(1-{1\over1+x^2}\right)dx+c\)
I = \(\rm {1\over2}\left[x^2\tan^{-1}x - (x-\tan^{-1}x)\right]+c\)
I = \(\boldsymbol{\rm {1\over2}\left[(x^2+1)\tan^{-1}x - x\right]+c}\)
