Integral of e^x(cosx)?

Homework Statement

I attached a picture of my attempt, it seems to loop back... Maybe I made a mistake... If not how am I suppose to integrate this? Thank you!

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Answers and Replies

Dick
Science Advisor
Homework Helper
Stop at the third line. Now move the integral of cos(x)*e^x on the right to the left. Then you are basically done. The third line gives you an equation where the only integral is cos(x)*e^x. Solve for it.

Stop at the third line. Now move the integral of cos(x)*e^x on the right to the left. Then you are basically done. The third line gives you an equation where the only integral is cos(x)*e^x. Solve for it.

Oh I see, I got

∫(e^xcosx dx)=(e^xsinx+e^xcosx)/2

But how did I get :
∫(e^xcosx dx)=e^xsinx+e^xcosx-∫(e^xcosx dx) (in the third step)

and

∫(e^xcosx dx)=∫(e^xcosx dx) (in the last step)

The two are not the same thing?

Dick
Science Advisor
Homework Helper
Oh I see, I got

∫(e^xcosx dx)=(e^xsinx+e^xcosx)/2

But how did I get :
∫(e^xcosx dx)=e^xsinx+e^xcosx-∫(e^xcosx dx) (in the third step)

and

∫(e^xcosx dx)=∫(e^xcosx dx) (in the last step)

The two are not the same thing?

Both of those statements are true. The first one tells you something useful. The second one is also true. But it doesn't tell you anything useful. They don't conflict with each other.

Last edited:
there is a really slick way to do this with Eulers formula. using e^(ix)=isin(x)+cos(x)
By substituting e^(ix) in and then taking the real part at the end.