Evaluate the following integrals: ^ 2x dx - Vidyadip

Question

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

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Answer

Let $\text{I}=\int\text{e}^{2\text{x}}\sin\text{x}\cos\text{x }\text{dx}$
$=\frac{1}{2}\int\text{e}^{2\text{x}}2\sin\text{x}\cos\text{x dx}$
$=\frac{1}{2}\int\text{e}^{2\text{x}}\sin2\text{x dx}$
We know that
$\int\text{e}^{2\text{x}}\sin\text{bx dx}=\frac{\text{e}^{2\text{x}}}{\text{a}^2+\text{b}^2}\{\text{a}\sin\text{bx}-\text{b}\cos\text{bx}\}+\text{C}$
$\Rightarrow\int\text{e}^{2\text{x}}\sin\text{2x dx}=\frac{\text{e}^{2\text{x}}}{8}\{2\sin2\text{x}-2\cos2\text{x}\}+\text{C}$
$\therefore\ \text{I}=\frac{1}{2}.\frac{\text{e}^{2\text{x}}}{8}\{2\sin2\text{x}-2\cos2\text{x}\}+\text{C}$
$\therefore\ \text{I}=\frac{\text{e}^{2\text{x}}}{8}\{\sin2\text{x}-\cos2\text{x}\}+\text{C}$

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