Expert: Paul Klarreich - 9/7/2009

Question
I have the function x^4(3x^5+4)^7, and need to find the indefinite integral of this function. I start by making a u substitution where 3x^5+4=u, and du/(15x^4)=dx. I replace dx and x with these two substitutions, which makes the integrand (1/15)u^7. After solving through I end up with (1/120)(3x^5+4)^8+c, but this is not the correct answer, any help would be appreciated.

Answer
Questioner: Alec
Country: United States
Category: Calculus
Private: No
Subject: An indefinite integration
Question: I have the function x^4(3x^5+4)^7, and need to find the indefinite integral of this function. I start by making a u substitution where 3x^5+4=u, and du/(15x^4)=dx.
I replace dx and x with these two substitutions, which makes the integrand (1/15)u^7. After solving through I end up with (1/120)(3x^5+4)^8+c, but this is not the correct answer, any help would be appreciated.
...............................................
{
| x^4 (3x^5 + 4)^7 dx
}

u = 3x^5 + 4,  du = 15x^4 dx
dx = du/15x^4

{
| x^4 (3x^5 + 4)^7 du/15x^4
}

{
| u^7 du/15 =
}

1/15 u^8/8 = u^8/120

(3x^5+4)^8/120 + c

Yes, it looks ok to me.  What answer did the book have?  Sometimes the book answer just looks different.  

You can also try the INTEGRATOR at:

http://integrals.wolfram.com/index.jsp

It can do lots of integrals.  In this case it multiplies out and integrates the polynomial.  It is obviously very powerful and also very stupid.