Integral help, how to approach this?

Question

anonymous · Answer

$$\int\limits_{0}^{1} \sin(y)(e-e^{y^{1/3} })dy$$ is the integral. I can plug it into wolfram alpha and get a value, but I would like to know what steps get me there>

abb0t · Answer

Wow. That's an interesting integral. I would suggest distributing the sine function across and take the integral sepearately. 
e is constant so you can just factor that out.
For the second integral, it gets a bit difficult. You could try and work with integration by parts twice.

anonymous · Answer

Except that the product of siny and e^y^1/3 is elliptic.  Neither of those will differentiate to zero.

anonymous · Answer

Heh, I have a whole homework sheet of these iterated integrals that the inner integral isnt bad, but the insides are just impossible. WA wont even give steps, just values when I include the bounds.. Arg

anonymous · Answer

That is one hairy integral!

anonymous · Answer

have you tried considering y^(1/3) as u ?

abb0t · Answer

I don't think you can use u-sub, since you don't have a du to substitute. But, I think since
the integral is from 0<y<1 so it might be easier to just use the bounds given to evaluate. In theory, I think you can use series to solve this.

anonymous · Answer

i would use taylor series to solve this problem

abb0t · Answer

Not series to solve the whole integral! Omg.

anonymous · Answer

i used taylor's expansion and i got 0.1644

anonymous · Answer

unless u want to use integration by parts and the Jacobin and polar coordinate,

anonymous · Answer

to solve this problem

abb0t · Answer

Well, depending on the course this is being taught in, which is probably a Calc BC (II) course, I would suggest parts. However, if this was for ODE, then I might suggest using poolar coordinates or series expansion.