Hi, I thought I had this question until I got to the very last part, I believe this question uses the repeated integration by parts method, is that right?

Evaluate the indefinite integral:

∫e^7x*sin(4x)

My work:

u = e^7x

u' = (1/7)e^7x

v = (-1/4)cos(4x)

v' = sin(4x)

∫e^7x*sin(4x) = (-1/4)e^7x*cos(4x) - ∫(-1/4)cos(4x)*(1/7)e^7x

.....................= (-1/4)e^7x*cos(4x) + (1/28) ∫cos(4x)*e^7x <-- integrate this indefinite integral using integration by parts?

So...

(1/28) ∫cos(4x)*e^7x

u = cos(4x)

u' = (-1/4)sin(4x)

v = (1/7)e^7x

v' = e^7x

And then I get...

(1/28) ∫cos(4x)*e^7x = (1/28)(cos(4x)*(1/7)e^7x + ∫(1/7)e^7x*(-1/4)sin(4x))

..............................= (1/196)(cos(4x) + (1/28^2)∫e^7x*sin(4x) <-- so this second term is now the same as the original equation, except for the constant of course, and I should just simplify, right? But I don't know the fractions make it look really complicated and I don't think I'm doing it right...

Does someone know where I went wrong and how to solve it properly?