Question

Calculate the following integral : \[\int\limits e^\left( ax \right) \cos(bx) dx\]

Answer 1

integration by parts.

Answer 2

u = e^{ax}
du / dx = ae^{ax}

dv = cos(bx)
v = bsin(bx)

Answer 3

\[= e^{ax}{bsin(bx) - \int bsin(bx)*ae^{ax}\]

Answer 4

\[ = e^{ax}{bsin(bx)} - \int bsin(bx)*ae^{ax}\]

Answer 5

integration by parts again.

Answer 6

\[ = e^{ax} bsin(bx) - ab \int sin(bx)e^{ax}\]

Answer 7

That's what I did, but won't that just go on infinitely?

Answer 8

then you add two integrals together.

Answer 9

oh, how do you do that?

Answer 10

hmm..what do you get when integrate again?
it'll be easier,

Answer 11

\[(sinbx*e^\left( ax \right))/a - \int\limits (1/a *e^\left( ax \right) * bcosbx) dx\]

Answer 12

That's just the new integral, so you'd then have to multiply that by the ab and put
b*e^(ax)∗sinbx
- in front as wel

Answer 13

give me a sec, i did it on paper, i'll upload it.

Answer 14

ok, great

Answer 15

there's no guarantee, i did it correctly.

Answer 16

and ye zoom in.

Answer 17

control +

Answer 18

Thank you, I think I understand the method, looks good to me!

Answer 19

:)