# Thread: (URGENT) integration by parts

1. ## (URGENT) integration by parts

let I = integral e^(ax) sin (bx) dx
use integration by parts to show that

I = e^(ax) * (a sin (bx) - b cos (bx)) / (a^2 + b^2) + C

i have tried every possible combination of two integration by parts and have not gotten the answer provided

can someone show me a detailed solution
help would be greatly appreciated

2. $I = \int e^{ax} \sin (bx) \ dx$

Use integration by parts: $u = e^{ax}$ and $dv = \sin bx dx$

To get: $I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}{\color{red}\int e^{ax}\cos (bx) dx}$

Now apply parts again to the integral in red using: $u = e^{ax}$ and $dv = \cos (bx) dx$

To get: $I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}\left({\color{red}\frac{1}{b}e^{ax} \sin (bx) - \frac{a}{b} \! \! \! \! \underbrace{\int e^{ax} \sin bx dx}_{{\color{black}\text{Doesn't this look familiar?}}}}\right)$

So we really have: $I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}\left(\frac{1}{b}e^{ax} \sin (bx) - \frac{a}{b}I\right)$

Now solve for $I$.

3. Originally Posted by razorfever
let I = integral e^(ax) sin (bx) dx
use integration by parts to show that

I = e^(ax) * (a sin (bx) - b cos (bx)) / (a^2 + b^2) + C

i have tried every possible combination of two integration by parts and have not gotten the answer provided

can someone show me a detailed solution
help would be greatly appreciated
this integral, and several like it, has been done many times on this forum, in many different ways.

see here for example