Evaluate the following definite integrals: _ 0 ^1 2x+3 5x^2+1 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\frac{2\text{x}+3}{5\text{x}^2+1}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{2\text{x}+3}{5\text{x}^2+1}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{1}\frac{2\text{x}+3}{5\text{x}^2+1}\text{ dx}+\int_{0}^\limits{1}\frac{3}{5\text{x}^2+1}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{5}\int_{0}^\limits{1}\frac{10\text{x}}{5\text{x}^2+1}\text{ dx}+3\int_{0}^\limits{1}\frac{1}{(\sqrt{5}\text{x})^2+1}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{5}\Big[\log\big(5\text{x}^2+1\big)\Big]^1_0+\frac{3}{\sqrt{5}}\Big[\tan^{-1}\big(\sqrt{5}\text{x}\big)\Big]^1_0$
$\Rightarrow\text{I}=\frac{1}{5}\log6+\frac{3}{\sqrt{5}}\tan^{-1}\sqrt{5}$

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