We have, I=∫3x^21 + x^6dx I=∫3x^21 + (x^3)^2dx Let t=x^3 dt=3x^2 dx Therefore, I=∫dt1 + t^2 I=tan^ 1t+C I=tan^ 1x^3+C Hence, thi ...

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Integration by Substitution

∫3x21 + x6dx

Question

$\int \frac{3 x^{2}}{1 + x^{6}} d x$

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Solution

We have,
$I = \int \frac{3 x^{2}}{1 + x^{6}} d x$

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

Let
$t = x^{3}$
$d t = 3 x^{2} d x$

Therefore,
$I = \int \frac{d t}{1 + t^{2}}$

$I = {tan}^{- 1} t + C$

$I = {tan}^{- 1} x^{3} + C$

Hence, this is the answer.

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