Evaluate the following integrals: e^x 16-e^ 2x dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{e}^\text{x}}{\sqrt{16-\text{e}^{2\text{x}}}}\text{ dx}$

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Answer

$\int\frac{\text{e}^\text{x}\text{dx}}{\sqrt{16-(\text{e}^{\text{x}})^2}}$

Let $\text{e}^\text{x}=\text{t}$

$\Rightarrow\text{e}^\text{x}\text{dx}=\text{dt}$

Now, $\int\frac{\text{e}^\text{x}\text{dx}}{\sqrt{16-(\text{e}^{\text{x}})^2}}$

$=\int\frac{\text{dt}}{\sqrt{16-\text{t}^2}}$

$=\int\frac{\text{dt}}{\sqrt{4^2-\text{t}^2}}$

$=\sin^{-1}\Big(\frac{\text{t}}{4}\Big)+\text{C}$

$=\sin^{-1}\Big(\frac{\text{e}^\text{x}}{4}\Big)+\text{C}$

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