Question

Integral (x^3)(e^(x^2))dx?

how to solve this using partial? please help with detail.

Answers

integral e^(x^2) x^3 dx

For the integrand e^(x^2) x^3, substitute u = x^2 and du = 2 x dx:

= 1/2 integral e^u u du

For the integrand e^u u, integrate by parts, integral f dg = f g- integral g df, where

f = u, dg = e^u du,

df = du, g = e^u:

= (e^u u)/2-1/2 integral e^u du

The integral of e^u is e^u:

= (e^u u)/2-e^u/2+constant

Substitute back for u = x^2:

= 1/2 e^(x^2) x^2-e^(x^2)/2+constant

Which is equal to:

= 1/2 e^(x^2) (x^2-1)+constant

#1

Let x^2 = u

=> 2x dx = du

=> integral

= (1/2) ∫ u e^u du

Integrating by parts

= (1/2) [u∫ e^u du - ∫ [d/du(u) ∫ e^u du]du]

= (1/2) (u e^u - e^u) + c

= (1/2) [x^2 e^(x^2) - e^(x^2)] + c

= (1/2) (x^2 - 1) e^(x^2) + c

#2

Make a substitution and then use integration by parts:

∫(x^3)*e^(x2) dx

u = x2

du/2 = x dx

1/2*∫u*e^(u) du

Using integration by parts on the ∫u*e^(u) du by letting z = u and dv = e^(u) du. The integral becomes:

1/2*[u*e^(u) - e^(u)] + C

e^(x2)/2 *(x2 - 1) + C

#3