Mathematics

Evaluate $$\int _ { 0 } ^ { 1 } \frac { 2 x + 3 } { 5 x ^ { 2 } + 1 } d x$$


SOLUTION

$$\\I=\int_{0}^{1}(\frac{2x}{5x^2+1})dx+\int_{0}^{1}(\frac{3}{5x^2+1})dx\\=(\frac{1}{5})\int_{0}^{1}(\frac{10x}{5x^2+1})dx+(\frac{3}{5})\int_{0}^{1}(\frac{1}{x^2+(\frac{1}{\sqrt{5}})^2})dx\\=(\frac{1}{5})\left[log(5x^2+1)\right]_{0}^{1}+(\frac{3}{5})(\frac{1}{(\frac{1}{\sqrt{5}})})\left[tan^{-1}(\frac{x}{(\frac{1}{\sqrt{5}})})\right]_{0}^{1}\\=(\frac{1}{5})log6+(\frac{3}{\sqrt{5}})tan^{-1}\sqrt{5}$$

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Subjective Medium Published on 17th 09, 2020
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