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

Question

Evaluate the following integrals:

$\int\frac{\sec^2\text{x}}{\sqrt{4+\tan^2\text{x}}}\text{ dx}$

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Answer

Let $\tan\text{x}=\text{t}$

$\Rightarrow\sec^2\text{x}\text{ dx}=\text{dt}$

$\Rightarrow\int\frac{\sec^2\text{x}}{\sqrt{\tan^2\text{x}+4}}\text{ dx}=\int\frac{\text{dt}}{\sqrt{\text{t}^2+2^2}}$

$=\log\Big|\text{t}+\sqrt{\text{t}^2+4}\Big|+\text{C}$

$=\log\Big|\tan+\sqrt{\tan^2\text{x}+4}\Big|+\text{C}$

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