Question

integral of e^(-x) cos8x dx

Answer 1

Use integration by parts.

Answer 2

I did and this is what I got: ∫ e⁽⁻ˣ⁾cos8xdx= (‑1/8)e⁽⁻ˣ⁾sin8x+ (1/8)[(1/8)e⁽⁻ˣ⁾cos8x+∫ e⁽⁻ˣ⁾cos8xdx

Answer 3

I am too lazy to write down the step by step detail, BUT, seeing this problem, I can see that you have to integrate twice. Once you integrate the second time, you are going to notice that you have \[\int\limits e^{-x} \cos(8x)dx = e^{-x}\cos(8x)+e^{-x}\frac{ 1 }{ 8 } \sin(x)-\int\limits e^{-x}\cos(x)dx\]
I think I did it right.You can double check my work. I did it in my head. But if you notice, you are just going to be going aorund in circles! But, notice that you have the same integral on both sides. Awesome! That means you can add them together! To get:\[2 \int\limits e^{-x}\cos(x)dx = e^{-x}\cos(8x)+e^{-x}\frac{ 1 }{ 8 }\sin(8x)\]
Now, all you have to do is divide by 2 and you are done!