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

Question

Evaluate the following integrals:
$\int\text{e}^{\cos^2\text{x}}\sin2\text{x}\text{ dx}$

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Answer

Let $\text{I}=\int\text{e}^{\cos^2\text{x}}\sin2\text{x}\text{ dx}$
Let $\cos^2\text{x}=\text{t}$
On differentiating both sides, we get
$-2\cos\text{x}\sin\text{x}\text{ dx}=\text{dt}$
$\therefore\text{I}=\int\text{e}^\text{t}2\sin\text{x}\cos\text{x}\frac{\text{dt}}{-2\sin\text{x}\cos\text{x}}$
$=-\int\text{e}^\text{t}\text{dt}$
$=-\text{e}^\text{t}+\text{C}$
$=-\text{e}^{\cos^2\text{x}}+\text{C}$

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