I=∫xx^2+1dx Let x^2+1=t 2xdx=dt xdx=12dt I=∫12dtt =12ln|t|+c I=12ln|x^2+1|+c . ...

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Integration by Substitution

Integrate ∫...

Question

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

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Solution

$I = \int \frac{x}{x^{2} + 1} d x$
Let $x^{2} + 1 = t$
$2 x d x = d t$
$x d x = \frac{1}{2} d t$
$I = \int \frac{1}{2} \frac{d t}{t}$
$= \frac{1}{2} l n | t | + c$
$I = \frac{1}{2} l n | x^{2} + 1 | + c$ .

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