Integration by Parts of x^5cos(x^3)

1. The problem statement, all variables and given/known data

[tex]\int x^5cos(x^3) dx[/tex]

2. Relevant equations
[tex]\int uv' = uv - \int u'v[/tex]


3. The attempt at a solution
[tex]\int x^5cos(x^3) dx[/tex]

[tex]\frac{(x^5)*(sin(x^3)}{(3x^2)} - \int\frac{5x^4*sin(x^3)}{(3x^2)} dx[/tex]

[tex]\frac{(x^3)*sin(x^3)}{3} - \int\frac{(5x^2)*sin(x^3)}{(3)} dx[/tex]

[tex]\frac{(x^3)*sin(x^3)}{3} - \frac{5}{3} \int x^2*sin(x^3) dx[/tex]

[tex]\int x^2*sin(u) *\frac{du}{3x^2}[/tex]

[tex]\frac{1}{3} \int sin(u) du[/tex]

[tex]\frac{-1}{3} cos(u)[/tex]

[tex]\frac{-1}{3} cos(x^3)[/tex]

Revisting

[tex]\frac{(x^3)*sin(x^3)}{3} - \frac{5}{3} * \frac{-1}{3} cos(x^3)[/tex]

My answer:

[tex]\frac{(x^3)*sin(x^3)}{3} + \frac{5}{9} * cos(x^3) + C[/tex]

Correct Answer:

[tex]\frac{(x^3)*sin(x^3)}{3} + \frac{1}{3} * cos(x^3) + C[/tex]

 

May I ask what the derivative of sin(x^3)/3x^2 may be then?

Thanks, rewriting the function got me the right answer.