How do you solve this using integration by parts?

i) Since differential of x^3 is 3x² and x^5 can be expressed as (x^3)*(x²), it is better for substituting x^3. ii) So let x^3 = t; differentiating, 3x² dx = dt; Also, at x = 0, t = 0 and at x = 2, t = 8 Further, 3(x^5) = (x^3)(3x²) iii) Hence of the above, the integral splits into: ∫(x^3){e^(x^3)(3x²) dx in (x = 0 to 2) = ∫t*(e^t) dt in (t = 0 to 8) iv) Integrating by parts, = [t*(e^t) - ∫(e^t) dt] in (t = 0 to 8) = [t*(e^t) - (e^t)] in (t = 0 to 8) Evaluating for the upper limit 8 and lower limit 0, = 8*(e^8) - (e^8) + 1 = 7*(e^8) + 1

Your using the wrong values for u and dv. You should be using dv = (3x^2)*(e^x^3)dx, since that would lead to v = (e^x^3). You can't integrate dv=e^x^3 dx because that doesn't take into account the (x^3) as an exponent. So, u = x^3 and du = 3x^2 dx, and dv = (3x^2)*(e^x^3)dx and v = e^x^3. It shouldn't be very difficult to do from there.