Calculus of a Single Variable, Chapter 8, 8.2, Section 8.2, Problem 56

Recall that indefinite integral follows int f(x) dx = F(x) +C where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as the constant of integration.
For the given integral problem: int 2x^3 cos(x^2) dx , we may apply apply u-substitution by letting: u = x^2 then du =2x dx .
Note that x^3 =x^2 *x then 2x^3 dx = 2*x^2 *x dx or x^2 * 2x dx
The integral becomes:
int 2x^3 cos(x^2) dx =int x^2 *cos(x^2) *2x dx
= int u cos(u) du
Apply formula of integration by parts: int f*g'=f*g - int g*f' .
Let: f =u then f' =du
g' =cos(u) du then g=sin(u)
Note: From the table of integrals, we have int cos(x) dx =sin(x) +C .
int u *cos(u) du = u*sin(u) -int sin(u) du
= usin(u) -(-cos(u)) +C
= usin(u) + cos(u)+C
Plug-in u = x^2 on usin(u) + cos(u)+C , we get the complete indefinite integral as:
int 2x^3 cos(x^2) dx =x^2sin(x^2) +cos(x^2) +C