integral question?

Question

integrate: sin(x^(1/3))dx

thx

Answer 1

Let t = x^(1/3), or x = t^3
[integral] sin(x^(1/3))dx
= [integral] 3t^2 sin t dt
= [integral] 3t^2 d(-cos t)
= - 3t^2cos t + [integral] 6t cos t dt
= - 3t^2cos t + [integral] 6t d(sin t)
= - 3t^2cos t + 6t sin t - [integral] 6sin t dt
= - 3t^2cos t + 6t sin t + 6cos t
= - 3x^(2/3)cos(x^(1/3)) + 6x^(1/3)sin(x^(1/3)) + 6cos(x^(1/3))