Question

$\int e^{2x}\left(\sin \left(ax+b\right)\right)dx$

Answer 1

$1=\int e_{ii}^{2x}\left(\sin ax+b\right)dx$

$=\sin (ax+b)\frac{e^{2x}}{2}-\frac{a}{2}\left[\cos (ax+b)\frac{e^{2x}}{2}+\int a\sin (ax+b)\frac{e^{2x}}{2}dx\right]$

$=\frac{e^{2x}}{2}\sin (ax+b)-\frac{a}{4}{e}^{2x}\cos (ax+b)-\frac{a^2}{4}\int e^{2x}\sin (ax+b)dx$

$1+\frac{a^2}{4}I=\frac{e^{2x}}{4}\left(2\sin (ax+b)-a\cos (ax+6)\right)$

$\left(\frac{4t{a}^2}{4}\right)I=\frac{e^{2x}}{4}\left(2\sin (ax+b)-a\cos (ax+b)\right)$

$I=\frac{e^{2}x}{a^2+4}\left(2\sin (ax+b)-a\cos (ax+b)+c\right)$