Evaluate the following integrals: ^ 2x (3x+4)dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\cos(3\text{x}+4)\text{dx}$

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Answer

Let $\text{I}=\int\text{e}^{2\text{x}}\cos(3\text{x}+4)\text{dx}$
Integrating by parts
$\text{I}=\text{e}^{2\text{x}}\frac{\sin(3\text{x}+4)}{3}-\int2\text{e}^{2\text{x}}\frac{\sin(3\text{x}+4)}{3}\text{dx}$
$=\frac{1}{3}\text{e}^{2\text{x}}\sin(3\text{x}+4)-\frac{2}{3}\int\text{e}^{2\text{x}}\sin(3\text{x}+4)\text{dx}$
$=\frac{1}{3}\text{e}^{2\text{x}}\sin(3\text{x}+4)-\frac{2}{3}\Big[-\text{e}^{2\text{x}}\frac{\cos(3\text{x}+4)}{3}+\int2\text{e}^{2\text{x}}\frac{\cos(3\text{x}+4)}{3}\text{dx}\Big]+\text{C}$
$\text{I}=\frac{\text{e}^{2\text{x}}}{9}\{2\cos(3\text{x}+4)+3\sin(3\text{x}+4)\}+\text{C}$
Hence,
$\text{I}=\frac{\text{e}^{2\text{x}}}{13}\{2\cos(3\text{x}+4)+3\sin(3\text{x}+4)\}+\text{C}$

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