Please explain integration by parts y= x^(3)e^(x^2)

Hello,

∫ x³ e^(x²) dx =

make a substitution, first; rewrite the integral as:

∫ x² e^(x²) x dx =

let:

x² = t

differentiate both sides:

d(x²) = dt

2x dx = dt

x dx = (1/2) dt

then, substituting:

∫ x² e^(x²) x dx = ∫ t e^t (1/2) dt =

integrate this by parts, letting:

e^t dt = dv → e^t = v

(1/2)t = u → (1/2) dt = du

yielding:

∫ u dv = u v - ∫ v du

∫ (1/2)t e^t dt = (1/2)t e^t - ∫ e^t (1/2) dt =

(1/2)t e^t - ∫ (1/2) e^t dt =

(1/2)t e^t - (1/2)e^t + C =

(1/2)e^t (t - 1) + C

substitute back x² for u, ending with:


∫ x³ e^(x²) dx = (1/2)e^(x²) (x² - 1) + C


I hope it's helpful