Evaluate the following integral: x^2-1 x^2+4 dx - Vidyadip

Question

Evaluate the following integral:
$\int\frac{\text{x}^2-1}{\text{x}^2+4}\text{ dx}$

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Answer

$\int\frac{\text{x}^2-1}{\text{x}^2+4}\text{ dx}$
$=\int\Big(\frac{\text{x}^2+4-4-1}{\text{x}^2+4}\Big)\text{ dx}$
$=\int\Big(\frac{\text{x}^2+4}{\text{x}^2+4}\Big)\text{ dx}-5\int\frac{\text{dx}}{\text{x}^2+2^2}$
$=\int\text{dx}-5\int\frac{\text{dx}}{\text{x}^2+2^2}$
$=\text{x}-\frac{5}{2}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$$\Big[\therefore\int\frac{\text{dx}}{\text{x}^2+\text{a}^2}=\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$

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