Integrating e^x(cosx) | Step-by-Step Solution for Homework

[theBEAST]

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!

 

[Dick]

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.

 

[theBEAST]

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]

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.

 

[cragar]

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.