Indefinite Integration Problem

**Sucker Punch** · October 19th 2010, 10:52 AM

I'm not sure how to approach this...

$\int e^{2x}sin(3x)$

I don't know where to start. This is on an assignment largely based on integration by parts, but I can't figure out what to do. If I do integration by parts, I still have to integrate

$\int e^{2x}cos(3x)$

so I'm not getting any closer to an answer.

Using online solving tools either gets me no answer, or something like this;

$\dfrac{e^{2x}(2sin(3x) - 3cos(3x))}{13}$

But it isn't much of a hint. Can someone point me in the right direction?

**tonio** · October 19th 2010, 12:04 PM

Originally Posted by Sucker Punch

I'm not sure how to approach this...

$\int e^{2x}sin(3x)$

I don't know where to start. This is on an assignment largely based on integration by parts, but I can't figure out what to do. If I do integration by parts, I still have to integrate

$\int e^{2x}cos(3x)$

so I'm not getting any closer to an answer.

Using online solving tools either gets me no answer, or something like this;

$\dfrac{e^{2x}(2sin(3x) - 3cos(3x))}{13}$

But it isn't much of a hint. Can someone point me in the right direction?

Do integration by parts TWICE, choosing $v'=e^{2x}$ in both cases, and then pass a summand from the RHS to the LHS...and voila!

Tonio

**Sucker Punch** · October 19th 2010, 12:39 PM

Originally Posted by tonio

Do integration by parts TWICE, choosing $v'=e^{2x}$ in both cases, and then pass a summand from the RHS to the LHS...and voila!

Tonio

Ah, of course. Thank you very much, I got it.

**TheCoffeeMachine** · October 19th 2010, 12:52 PM

Then there is the formula:

$\displaystyle \int e^{ax}\sin(bx) dx = e^{ax}\left\{\frac{a\sin{bx}-b\cos{bx}}{a^2+b^2}\right\}+k$

Similarly:

$\displaystyle \int e^{ax}\cos(bx) dx = e^{ax}\left\{\frac{a\cos{bx}+b\sin{bx}}{a^2+b^2}\r ight\}+k$

They can be derived using by-parts twice -- same as the problem.

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