7.1 Integration By Parts/17

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int\left(e^{2{\theta}}\right)sin{3\theta} d{\theta}}

First:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u={\sin{(3\theta)}}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle du={3\cos{(3\theta)}} d{\theta}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dv=\left(e^{2{\theta}}\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v=\frac{1}{2}\left(e^{2{\theta}}\right)}

Then: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U={\cos{(3\theta)}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dU={3\sin{(3\theta)}} d{\theta}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dV=\left(e^{2{\theta}}\right) d{\theta}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=\frac{1}{2}\left(e^{2{\theta}}\right)} to get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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,$I={\frac {1}{13}}\left(e^{2{\theta }}\right)2sin{3\theta }-{3cos3\theta }+C$,

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C=\frac{4}{13}C}