Question

∫[0,π,e^(3t)sin(3t),]dt=

Answer 1

have u try using substitution?

Answer 2

You should use integration by parts:\[\int\limits_{}{}u.dv=uv-\int\limits_{}{}v.du\]and you'll have to use it a couple of times. Set the following:\[u=e^{3t}\]and\[dv=\sin(3t)dt\]then \[du=3e^{3t}dt\]and\[v=-\frac{1}{3}\cos(3t)\]Then your integral becomes\[\int\limits_{}{}e^{3t}\sin(3t)dt=-\frac{1}{3}e^{3t}\cos(3t)+\int\limits_{}{}\cos(3t)e^{3t}dt\]

Answer 3

Now you have to repeat the process on the final integral. Set u=e^(3t) and dv=cos(3t)dt and repeat. You should get\[\int\limits_{}{}e^{3t}\cos(3t)dt=\frac{1}{3}e^{3t}\sin(3t)-\int\limits_{}{}e^{3t}\sin(3t)dt\]

Answer 4

Substitute this into the equation before it, so that\[\int\limits_{}{}e^{3t}\sin(3t)dt=-\frac{1}{3}e^{3t}\cos(3t)\]\[+\left[ \frac{1}{3}e^{3t}\sin(3t)-\int\limits_{}{}e^{3t}\sin(3t)dt \right]\]

Answer 5

Add \[\int\limits_{}{}e^{3t}\sin(3t)dt\]to both sides. Then\[2\int\limits_{}{}e^{3t}\sin(3t)dt=\frac{1}{3}e^{3t}\left( \sin(3t)-\cos(3t) \right)\]

Answer 6

Divide both sides by 2 to get your integral:\[\int\limits_{}{}e^{3t}\sin(3t)=\frac{1}{6}e^{3t}\left( \sin(3t)-\cos(3t) \right)+C\]

Answer 7

There is a method by which you can pump out IBP problems extremely quickly, but it's hard to explain online...so I stuck to the 'traditional' route.