Question

how do i integrate sqrt(1+x^3)??

Answer 1

If you are a student, you aren't expected to be able to do this without Wolfram|Alpha. Seriously: http://www.wolframalpha.com/input/?i=integrate+sqrt%281%2Bx%5E3%29

Answer 2

put 1+x^3=t^2
substitute t for x

Answer 3

How does that help? With that, we get: \[ 1 + x^3 = t^2 \Rightarrow 3x^2 = 2t \cdot \dfrac {\mathrm{d}t}{\mathrm{d}x} \Rightarrow \mathrm{d}x = \dfrac {2t}{3x^2} \, \mathrm{d}t \]\[ \int \sqrt{1+x^3} \, \mathrm{d}x = \int \sqrt {t^2} \cdot \dfrac {2t}{3x^2} \, \mathrm{d}t = \int \dfrac {2t^2}{3x^2} \, \mathrm{d}t \]Since \[ x^3 = t^2 - 1, \]we have: \[ x^2 = (t^2 - 1)^{2/3}, \]so we go back to this integral (which is more like a punishment =P): \[ \int \dfrac {2t^2}{3\sqrt[3]{(t^2 - 1)^2}} \]Do you really even want to attempt integrating this?