I was doing Integration by parts. I found an example which looks like :
$$e^{ax}\cos (bx+c)\mathrm dx$$
While I was integrating it by parts. I noticed it is repeating. When it repeated first moment my book wrote $I$ instead of above equation where $I=e^{ax}\cos (bx+c)\mathrm dx$. The same value is repeating all the time that's why my book didn't work further. They had did something else. They had found an equation for $I\cdot a^2$. Another equation for $I(a^2+b^2)$. Then, they had sent $a^2+b^2$ to RHS. Then, they had took $a=r\sin \theta$, $b=r\cos\theta$, $r^2=a^2+b^2$ and, $\tan\theta=\frac{b}{a}$ or, $\theta=\tan^{-1}\frac{b}{a}$. Then, they had put these values in those earlier equations(which derived from first equation(which I wrote at first or, in title) ).
$$\int uv\mathrm dx=u\int vdx-\int[\mathrm {\frac{du}{dx}}\cdot \int v\mathrm dx]\mathrm dx$$
They took $u=\cos (bx+c)$ and, $v=e^{ax}$
- Could you please describe integration by parts to me? (I can solve normal problems but, when it came to this one it was little bit looking harder to me. Even, I wasn't able to understand integration of it.)
- Why they had took (found) different equations?
If steps are needed than, I can add steps also..
