How would you solve this integral?

Question

I am trying to solve this integral $$\int_1^{2}\int_\frac{1}{x}^{x^2}(y^2-x) dydx$$ However I don't get the right answer. I'm not showing what I have tried because I just want guidance on how to integrate the $dy$ part.

Answer 1

The way to think about this is just to think of this problem in parts:

(1) $\int_1^{2}\underbrace{\int_\frac{1}{x}^{x^2}(y^2-x) dy}_{\text{you treat x like a constant and y like a variable here for now}}dx$.

(2) So then you get $\int_1^{2}\underbrace{((\frac{y^3}{3}-xy)]_{\frac{1}{x}}^{x^2}}_{\text{you then do your definite integral part just replacing these values for y}}dx$.

(3) $\int_1^{2}[\frac{(x^2)^3}{3}-x(x^2)]-[\frac{(\frac{1}{x})^3}{3}-x(\frac{1}{x})]dx$

Then, you got it from there.