7.1 Integration By Parts/17

$\int \left(e^{2{\theta }}\right)sin{3\theta }d{\theta }$

First:

$u={\sin {(3\theta )}}$ $du={3\cos {(3\theta )}}d{\theta }$

$dv=\left(e^{2{\theta }}\right)$ Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v={\frac {1}{2}}\left(e^{2{\theta }}\right)}

Then: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I=\int \left(e^{2{\theta }}\right)\sin {3\theta }d{\theta }={\frac {1}{2}}\left(e^{2{\theta }}\right)*{\sin {(3\theta )}}-{\frac {3}{2}}\int \left(e^{2{\theta }}\right)\cos {3\theta }d{\theta }}

Next: let

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle U={\cos {(3\theta )}}}

$dU={3\sin {(3\theta )}}d{\theta }$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dV=\left(e^{2{\theta }}\right)d{\theta }}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V={\frac {1}{2}}\left(e^{2{\theta }}\right)} to get

$\int \left(e^{2{\theta }}\right)\cos {3\theta }d{\theta }={\frac {1}{2}}\int \left(e^{2{\theta }}\right)\cos {3\theta }d{\theta }+{\frac {3}{2}}\int \left(e^{2{\theta }}\right)\sin {3\theta }d{\theta }$ substituting in the previous formula gives

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I={\frac {1}{2}}\left(e^{2{\theta }}\right)\sin {3\theta }-{\frac {3}{4}}\left(e^{2{\theta }}\right)\cos {3\theta }-{\frac {9}{4}}\int \left(e^{2{\theta }}\right)sin{3\theta }d{\theta }}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ={\frac {1}{2}}\left(e^{2{\theta }}\right)sin{3\theta }-{\frac {3}{4}}\left(e^{2{\theta }}\right)cos{3\theta }-{\frac {9}{4}}I}

Then: ${\frac {13}{4}}I={\frac {1}{2}}\left(e^{2{\theta }}\right)sin{3\theta }-{\frac {3}{4}}\left(e^{2{\theta }}\right)cos{3\theta }+C$,

Hence,Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I={\frac {1}{13}}\left(e^{2{\theta }}\right)2sin{3\theta }-{3cos3\theta }+C} ,

Where Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C={\frac {4}{13}}C}