Indefinite Integral of e^(7x)(sin(4x))dx

cant seem to get the substitution/integration by parts to work for me

thanks in advance!

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- September 9th 2008, 06:58 PMcasesamcalculus integration problem
Indefinite Integral of e^(7x)(sin(4x))dx

cant seem to get the substitution/integration by parts to work for me

thanks in advance! - September 9th 2008, 07:09 PMChop Suey
Use integration by parts

and treat the integral as the unknown. Then you can solve for the integral as if it's an algebra problem.*twice* - September 9th 2008, 07:34 PMcasesam
- September 9th 2008, 09:25 PMProve It
Integration by parts should work, but sometimes one of the most powerful methods of integration is by taking derivatives.

What we are trying to find is

.

If we find the derivative of I, we get

In reverse, we get that

and with a bit of rearranging we find

With me so far?

Now, if we were to divide both sides of the equation by 7, we'd have another expression for I (the function we are integrating). But there's another integral... Let's use the same process to solve for this integral.

Let's call this integral J...

so if .

Taking the derivative of J, we get

or in reverse, that

.

With a bit of rearranging we find that...

or

Substituting J into our original equation gives...

which, when expanded and manipulated, reduces to

And finally, we can solve for I, which is what we were finding...

(the integration constant).

Yes, it is a fair bit of writing, but is extremely powerful if all else fails...