We have, #10; #10; I=∫xdxx^2+1 #10; #10; #160;Let t=x^2+1 #10; #10; dtdx=2x+0 #10; #10; dt2=xdx #10; #10; #160;Ther ...

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Integration by Substitution

Solve: ∫xdxx...

Question

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

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Solution

We have,

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

Let $t = x^{2} + 1$

$\frac{d t}{d x} = 2 x + 0$

$\frac{d t}{2} = x d x$

Therefore,

$I = \frac{1}{2} \int \frac{d t}{t}$

$I = \frac{1}{2} ln (t) + C$

On putting the value of $t$ , we get

$I = \frac{1}{2} ln (x^{2} + 1) + C$

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