What is the integral of $x^{-2\theta}$?

Question

What is the integral of $x^{-2\theta}$?

Not sure how the integration works with two variables. Can anyone confirm?

Answer 1

You have to choose a variable to integrate, the other remains as a constant. Or do both, one at a time.

Integrating with respect to x: \begin{align*} \int x^{-2 \theta} dx &= \frac{x^{1 - 2\theta}}{1 - 2\theta} + C\\ \end{align*} Integrating with respect to θ: \begin{align*} \int x^{-2 \theta} d \theta &= -\frac{x^{-2 \theta}}{2 \ln(x)} + C \end{align*}