Make substitutions for W & K to be able to use integral table

[leo255]

Homework Statement

[/B]
Find a substitution w and a constant k so that the integral x^5 e^(bx^2) dx can be written in the form kw^2 * e^(bw) dw, and evaluate the integral (answer may involve the constant b).

Homework Equations

Integral of x^2 e^bx dx = e^bx ((x^2 / b) - (2x/b^2) + (2/b^3) ) + C

The Attempt at a Solution

[/B]
w = x^2, dw = 2xdx
k = 1/2

I would get: 2 * the integral of w^2 * e^dw. Using k = 1/2, I can get rid of the two, and would have:

e^bw ((w^2 / b) - (wx/b^2) + (2/b^3) ) + C

e^(b*x^2) ((x^4 / b) - (2x^2/b^2) + (2/b^3) ) + C

Please let me know if and where I am off in my solution. Thanks.

 

[Mark44]

Homework Statement

[/B]
Find a substitution w and a constant k so that the integral x^5 e^(bx^2) dx can be written in the form kw^2 * e^(bw) dw, and evaluate the integral (answer may involve the constant b).

Homework Equations

Integral of x^2 e^bx dx = e^bx ((x^2 / b) - (2x/b^2) + (2/b^3) ) + C

The Attempt at a Solution

[/B]
w = x^2, dw = 2xdx
k = 1/2

I would get: 2 * the integral of w^2 * e^dw. Using k = 1/2, I can get rid of the two, and would have:

e^bw ((w^2 / b) - (wx/b^2) + (2/b^3) ) + C

e^(b*x^2) ((x^4 / b) - (2x^2/b^2) + (2/b^3) ) + C

Please let me know if and where I am off in my solution. Thanks.

You can find that out for yourself. If you differentiate your solution, you should get x⁵e^bx2.