How do you integrate #int x^3 e^(x^2 ) dx # using integration by parts?

Question

How do you integrate #int x^3  e^(x^2   ) dx  # using integration by parts?

Christopher Allers · Accepted Answer

#= 1/2e^(x^2) ( x^2 - 1 ) + C# throughout , remember that #d/dx(e^(x^2) )= 2x e^(x^2) # So: #int x^3 e^(x^2 ) dx# setting it up for IBP #= int color(red)(x^2) d/dx(color(blue)(1/2 e^(x^2 ))) dx# so by IBP #= color(red)(x^2) (color(blue)(1/2 e^(x^2 )) ) - int d/dx( color(red)(x^2)) (color(blue)(1/2 e^(x^2 ))) dx# #= 1/2 x^2 e^(x^2) - int 2 x * 1/2 e^(x^2 ) dx# #= 1/2 x^2 e^(x^2) - int x e^(x^2 ) dx# #= 1/2 x^2 e^(x^2) - int 1/2 d/dx( e^(x^2 )) dx# #= 1/2 x^2 e^(x^2) - 1/2 e^(x^2 ) + C# #= 1/2e^(x^2) ( x^2 - 1 ) + C#

Scarlett Allison · Answer

undefined To integrate ∫x^3 * e^(x^2) dx using integration by parts, we apply the integration by parts formula:

∫u dv = uv - ∫v du

Let u = x^3 and dv = e^(x^2) dx. Then, differentiate u to get du and integrate dv to get v.

du = 3x^2 dx
v = ∫e^(x^2) dx

To integrate v, we perform a substitution. Let w = x^2, then dw = 2x dx. Therefore, ∫e^(x^2) dx becomes ∫(1/2)e^w dw, which integrates to (1/2)e^(x^2).

Now, we can apply the integration by parts formula:

∫x^3 * e^(x^2) dx = x^3 * (1/2)e^(x^2) - ∫(1/2)e^(x^2) * 3x^2 dx

Simplify:

= (1/2)x^3 * e^(x^2) - (3/2)∫x^2 * e^(x^2) dx

We can apply integration by parts again to evaluate the remaining integral. Let u = x^2 and dv = e^(x^2) dx. Then, differentiate u to get du and integrate dv to get v.

du = 2x dx
v = (1/2)e^(x^2)

Now, we apply the integration by parts formula:

= (1/2)x^3 * e^(x^2) - (3/2)[x^2 * (1/2)e^(x^2) - ∫(1/2)e^(x^2) * 2x dx]

= (1/2)x^3 * e^(x^2) - (3/2)[(1/2)x^2 * e^(x^2) - ∫x * e^(x^2) dx]

We continue this process until we can integrate the remaining term. However, integrating xe^(x^2) requires another substitution.

Let z = x^2, then dz = 2x dx. Therefore, ∫xe^(x^2) dx becomes ∫(1/2)e^z dz, which integrates to (1/2)e^(x^2).

Substitute back into the equation and simplify to obtain the final result.