Evaluate the following integrals:


Evaluate the following integrals:

$\int e^{a x} \sin (b x+c) d x$


Let $I=\int e^{a x} \sin (b x+c) d x$

$=-e^{a x} \frac{\cos (b x+c)}{b}+\int a e^{a x} \frac{\cos (b x+c)}{b} d x$

$=-\frac{1}{b} e^{a x} \cos (b x+c)+\frac{a}{b} \int e^{a x} \cos (b x+c)$

$I=\frac{e^{a x}}{b^{2}}\{a \sin (b x+c)-b \cos (b x+c)\}-\frac{a^{2}}{b^{2}} I+c_{1}$

$\mathrm{I}=\left\{\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{~b}^{2}}\right\}-\frac{\mathrm{e}^{\mathrm{ax}}}{\mathrm{b}^{2}}\{\mathrm{a} \sin (\mathrm{bx}+\mathrm{c})-\mathrm{b} \cos (\mathrm{bx}+\mathrm{c})\}+\mathrm{c}_{1}$

$=\frac{e^{a x}}{a^{2}+b^{2}}\{a \sin (b x+c)-b \cos (b x+c)\}$

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