Evaluate the integral _ 0 ^ 1 x x^ 2 +1 d x using substitution. - Vidyadip

Question

Evaluate the integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ using substitution.

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Answer

Given integral is: $\int_{0}^{1} \frac{x}{x^{2}+1} d x $
Let $x^2 + 1 = t$
$\Rightarrow 2xdx = dt$
$\Rightarrow xdx = \frac{1}{2} dt$
When $x = 0, t = 1$ and when $x = 1, t = 2$
$\Rightarrow \int_{0}^{1} \frac{x}{x^{2}+1} d x=\int_{1}^{2} \frac{d t}{2 t} $
$=\frac{1}{2} \int_{1}^{2} \frac{d t}{t} $
$=\frac{1}{2}\left[\log |t| ]_{1}^{2}\right. $
$= \frac{1}{2}[\log 2-\log 1] $
$= \frac{1}{2} \log 2$

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