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

Question

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}(-\sin\text{x}+2\cos\text{x})\text{dx}$

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Answer

Let $\text{I}=\int\text{e}^{2\text{x}}(-\sin\text{x}+2\cos\text{x})\text{dx}$
$=-\int\text{e}^{2\text{x}}\sin\text{x dx}+2\int\text{e}^{2\text{x}}\cos\text{x dx}$
Applying by parts in the $2^{nd}$^ integrand
$\therefore\text{I}=-\int\text{e}^{2\text{x}}\sin\text{x dx}+2\Big\{\frac{1}{2}\text{e}^{2\text{x}}\cos\text{x}+\int\frac{1}{2}\text{e}^{2\text{x}}\sin\text{x dx}\Big\}$
$=-\int\text{e}^{2\text{x}}\sin\text{x dx}+\text{e}^{2\text{x}}\cos\text{x}+\int\text{e}^{2\text{x}}\sin\text{x dx}+\text{C}$
$=\text{e}^{2\text{x}}\cos\text{x}+\text{C}$
Thus,
$\text{I}=\text{e}^{2\text{x}}\cos\text{x + C}$

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