How to Integrate ∫x(2^x^3)dx Using Substitution and Logarithms

[rmiller70015]

Homework Statement

∫x(2^x^3)dx

Homework Equations

The Attempt at a Solution

I've tried using substitution using both x^3 and 2^x^3 as u.

I did get pretty far by using log_2 on each side.

∫log_2(x2^x^3)dx=∫(log_2(x)+log_2(2^x^3))dx=∫(log_2(x)+x^3)dx

At this point I'm not sure what to do, any help or hints would be appreciated.

 

[SteamKing]

See this thread for a hint:

https://www.physicsforums.com/showthread.php?t=734992

The OP there faced a similar integration problem.

 

[Dick]

Homework Statement

∫x(2^x^3)dx

Homework Equations

The Attempt at a Solution

I've tried using substitution using both x^3 and 2^x^3 as u.

I did get pretty far by using log_2 on each side.

∫log_2(x2^x^3)dx=∫(log_2(x)+log_2(2^x^3))dx=∫(log_2(x)+x^3)dx

At this point I'm not sure what to do, any help or hints would be appreciated.

Two points. i) you can't take log inside an integral and get anything having anything to do with the original integral. ii) 2^x^3 doesn't mean anything. You mean either (2^x)^3 or 2^(x^3). They are very different. I suspect you mean (2^x)^3. That you can do with some work and integration by parts. 2^(x^3) leads to a nonelementary integral.

 

[rmiller70015]

It was 2^(x^3) all multiplied by x.

 

[Dick]

If your integrand is $x 2^{(x^3)}$ as opposed to $x (2^x)^3$ then you need some species of a gamma function to get an integral. Have you talked about those?