Integration with Complex Numbers

[waealu]

Homework Statement

Evaluate ∫$^{∏}_{0}$e^(1+i)xdx

Homework Equations

I know that the Real part of this is -(1+e^∏)/2 and the Imaginary part is (1+e^∏)/2, but I can't get the right solution.

I tried using u-substitution to create something that looked like ∫$^{∏}_{0}$((e^u)/(1+i))du

but I don't think that's correct. How do I solve this? Thanks!

 

[Dick]

Homework Statement

Evaluate ∫$^{∏}_{0}$e^(1+i)xdx

Homework Equations

I know that the Real part of this is -(1+e^∏)/2 and the Imaginary part is (1+e^∏)/2, but I can't get the right solution.

I tried using u-substitution to create something that looked like ∫$^{∏}_{0}$((e^u)/(1+i))du

but I don't think that's correct. How do I solve this? Thanks!

Your substitution is fine. Except the u limits aren't the same as the x limits. Just do the u integration and then change back to x.

 

[HallsofIvy]

Or, equivalently, change the limits of integration as you change the variable. You made the substitution u= (1+ i)x so that du= (1+ i)dx so that dx= du/(1+ i). If $x= \pi$ then $u= (1+ i)\pi$ and if x= 0, u= 0 so the integral becomes
$$\frac{1}{1+ i}\int_0^{(1+i)\pi} e^udu$$