First slide

Evaluation of definite integrals

Question

$\int \frac{x^{2} \tan^{- 1} (x^{3})}{1 + x^{6}} d x$ =

Easy

A

$\frac{1}{3} {(\tan^{- 1} x)}^{3} + C$
B

$\frac{1}{6} {(\tan^{- 1} x^{3})}^{2} + C$
C

$\frac{1}{3} {(\tan^{- 1} x^{2})}^{3} + C$
D

$\frac{1}{3} {(\tan^{- 1} x^{3})}^{3} + C$

Solution

                  put    $\tan^{- 1} x^{3} = t$ $\Rightarrow \frac{1 (3 x^{2}) d x}{1 + x^{6}} = d t$

              $\int \frac{x^{2} \tan^{- 1} (x^{3})}{1 + x^{6}} d x$       =     $\frac{1}{3} \int t d t = \frac{t^{2}}{6} + C$
                                      $= \frac{1}{6} {(\tan^{- 1} x^{3})}^{2} + C$

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