Evaluate the following definite integrals: _ 0 ^1 x x+1 dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{1}\frac{\text{x}}{\text{x}+1}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{\text{x}}{\text{x}+1}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{1}1-\frac{1}{\text{x}+1}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x}-\log(\text{x}+1)\big]_0^1$
$\Rightarrow\text{I}=1-\log2-(0-\log1)$
$\Rightarrow\text{I}=\log\text{e}-\log2$
$\Rightarrow\text{I}=\log\frac{\text{e}}{2}$

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