For which value of a does int e^(ax) cos(x) dx converge?

[mat145]

Homework Statement

For which value the definite integral (0 to infinity) e^ax cos(x) dx converges? Calculate for the value of a.

Mathieu

 

[Saladsamurai]

You must show an attempt at a solution. Forum rules (and plus I'm lazy).
Now, how do you evaluate an improper integral?

Casey

 

[mat145]

It depends if it's a type 1 or 2. For type 1 you must calculate the integral by replacing the infinity by a variable and then calculate the limit using this variable. For type 2 with a discontinuity you must split the integral in two parts and then calculate the integral using the same way than for type 1.

Mathieu

 

[Gib Z]

It is the first type. So replace the infinity with a variable, and evaluate the integral.

 

[mat145]

ok the integral is:

(((a cos(x))/((a^2)+1)) + (sin(x)/((a^2)+1))) e^ax

but how do I calculate the limit?

Mathieu

 

[Gib Z]

I haven't calculated the indefinite integral myself, so I am not sure if yours is correct, but now your integral was the difference of that expression evaluated at b, as b goes to infinity, and that expression evaluated at 0.

 

[mat145]

I understand the b to infinity part but I do not understand the expression evaluated at 0 part.

Mathieu

 

[Gib Z]

The Fundamental theorem of calculus states roughly that $$\int^b_a f(x) dx = F(b) - F(a)$$ where F'(x) = f(x). So that evaluated at 0 part corresponds to the -F(a) part.