Question

Check my work on a tricky integral please?

Answer 1

\[\int\limits_{0}^{1} \int\limits_{0}^{x^3} e^{x}\sin(y)dydx\]

Answer 2

Now because dy is first, e^x I treated as a constant and pulled it out of the integral. This left me with just sin(y) to inegrate which is just -cos(y) so now Ive got:

\[-e^{x}\cos(y) \]

evaluating the integral from y=0 to y=x^3 I get:

\[-e^{x}\cos(x^{3})+e^{x}\]

Answer 3

As long as this looks ok I'll be happy, because this is the point that I got stuck. The outer integral is:
\[\int\limits_{0}^{1} -e^{x}\cos(x^{3})+e^{x}dx\]

And I am not sure where to go from here

Answer 4

This might be one of those situations where you need to change your order of integration.

Answer 5

I know I can separate it into pieces by puting the -e^xcos(x^2) to one side and e^x to the other.. but I dont know how to integrate (without cheating and using wolframalpha) \[-e^{x}\cos(x^{3})\]

Answer 6

Because of the cos(x^2) ?

Answer 7

Because you hit a dead end in that one direction.