# Double integral with two variables in the exponential: $\int_0^\pi \int_0^\infty e^{-ibx\cos\theta - {x/a}} x^2 \sin\theta ~dx~d\theta$.

I'm having trouble with the substitutions in the following integral:

$$\int_0^\pi \int_0^\infty e^{-ibx\cos\theta - {x/a}} x^2 \sin\theta ~dx~d\theta$$

My attempt:

Let $$u = \cos\theta$$ then $$du = -\sin\theta~d\theta$$

Then we have

$$-1 \int_0^\pi \int_0^\infty e^{-ibxu - {x/a}} x^2 ~du~dx$$

How do I separate the u out of the exponential to integrate separately with respect to $u$ and then $x$? Am I doing the wrong substitution?

• Seems you forgot to change the bounds after the substitution. – Simply Beautiful Art Dec 6 '17 at 2:26
• When you change variables, you have to change the endpoints to match: $\theta = 0$ corresponds to $u = \cos \theta = 1$, $\theta = \pi$ to $u = -1$. – Robert Israel Dec 6 '17 at 2:26
• ... and then you can do the $du$ integral, treating $x$ as constant. – Robert Israel Dec 6 '17 at 2:31
• Why do we treat x as constant? – Ella Dec 6 '17 at 2:33
• @Ella As $u$ changes, $\theta$ changes, but does $x$ change? – Simply Beautiful Art Dec 6 '17 at 2:43