`int 2x^3cos(x^2) dx` Find the indefinite integral by using substitution followed by integration by parts.

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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`

 

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