Evaluate the integral? : # int x^3e^(x^2) dx #

Question

Evaluate the integral? : # int  x^3e^(x^2) dx #

Isabella Ackerman · Accepted Answer

The answer is #=1/2e ^(x^2)(x^2-1) +C#

We need the integration by parts

#intp'qdx=pq-intpq'dx#

We perform the substitution

Let #u=x^2#, #=>#, #du=2xdx#

Therefore,

#intx^3e^(x^2)dx=1/2intue^ udu#

We apply the integration by parts

#p'(u)=e^u#, #=>#, #p(u)=e^u#

#q(u)=u#, #=>#, #q'(u)=1#

Therefore,

#1/2intue^ udu=1/2(ue^u-inte^udu)=1/2(ue^u-e ^u)#

#=1/2e ^(x^2)(x^2-1) +C#

Ezra Adams · Answer

We seek:

# int \ x^3e^(x^2) \ dx = 1/2(x^2-1)e^(x^2) + C #

We seek: # I = int \ x^3e^(x^2) \ dx # Note as a helper that: # d/dx ( e^(x^2) ) = 2xe^(x^2) iff int \ 2xe^(x^2) \ dx = e^(x^2) # So we can write the integral as: # 2I = int \ (x^2)(2xe^(x^2)) \ dx # We can now use the formula for Integration By Parts (IBP): # int \ u(dv)/dx \ dx = uv - int \ v(du)/dx \ dx #, or less formally # " " int \ u \ dv=uv-int \ v \ du # I was taught to remember the less formal rule in word; "The integral of udv equals uv minus the integral of vdu". If you struggle to remember the rule, then it may help to see that it comes a s a direct consequence of integrating the Product Rule for differentiation. Let # { (u,=x^2, => (du)/dx,=2x), ((dv)/dx,=2xe^(x^2), => v,=e^(x^2) ) :}# Then plugging into the IBP formula: # int \ (u)((dv)/dx) \ dx = (u)(v) - int \ (v)((du)/dx) \ dx # gives us # int \ (x^2)(2xe^(x^2)) \ dx = (x^2)(e^(x^2)) - int \ (e^(x^2))(2x) \ dx # # :. 2I = x^2e^(x^2) - int \ 2xe^(x^2) \ dx + A # # " " = x^2e^(x^2) - e^(x^2) + A # # " " = (x^2-1)e^(x^2) + A # Hence # I = 1/2(x^2-1)e^(x^2) + C #

Christopher Agresta · Answer

undefined To evaluate the integral ∫x^3e^(x^2) dx, you can use the technique of integration by parts. Let u = x^2 and dv = x^3e^(x^2) dx. Then, differentiate u to get du = 2x dx, and integrate dv to get v = (1/2)e^(x^2).

Now, apply the integration by parts formula:

∫udv = uv - ∫v du

Substitute the values of u, dv, du, and v into the formula:

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

Simplify:

= (1/2)x^2e^(x^2) - ∫xe^(x^2) dx

This integral can be solved by another substitution. Let w = x^2, then dw = 2x dx. This transforms the integral into:

∫xe^(x^2) dx = (1/2)∫e^w dw

Integrate:

= (1/2)e^w + C

Substitute w back in terms of x:

= (1/2)e^(x^2) + C

Now, put this result back into the first integral:

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

So, the integral of x^3e^(x^2) dx is (1/2)x^2e^(x^2) - (1/2)e^(x^2) + C, where C is the constant of integration.