# Thread: Exponential and Trig in One Integral

1. ## Exponential and Trig in One Integral

How do you integrate $\int e^{x}\cos{(2x)}dx$?

I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. Can anyone explain?

2. Originally Posted by Keithfert488
How do you integrate $\int e^{x}\cos{(2x)}dx$?

I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. Can anyone explain?
move that "repeated" integral to the side of the equation with the original integral.

3. Originally Posted by Keithfert488
How do you integrate $\int e^{x}\cos{(2x)}dx$?

I started to try integration by parts but I noticed quite quickly that it would not work because I would continue to get the same integral. <<<< That's the trick!
Can anyone explain?
Use integration by parts:

First step:

$\int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+\int2e^x \cdot \sin(2x)dx$

Second step:

$\int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+2e^x \cdot \sin(2x)- \int 4e^x\cdot \cos(2x)dx$

Now collect the integrals at the LHS:

$5\int e^{x}\cos{(2x)}dx = e^x \cdot \cos(2x)+2e^x \cdot \sin(2x)$

Therefore:

$\int e^{x}\cos{(2x)}dx = \frac15 e^x \cdot (\cos(2x)+2 \sin(2x))$

EDIT: ...as usual: Too late

4. Oh I see now. Thanks to both of you.