Integral of x^3cosx dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 3 x^{2}$.

   To find $v{\left(x \right)}$:

   1. The integral of cosine is sine:

      $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

   Now evaluate the sub-integral.

2. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = 3 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 6 x$.

   To find $v{\left(x \right)}$:

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

   Now evaluate the sub-integral.

3. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = - 6 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = -6$.

   To find $v{\left(x \right)}$:

   1. The integral of cosine is sine:

      $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

   Now evaluate the sub-integral.

4. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \left(- 6 \sin{\left(x \right)}\right)\, dx = - 6 \int \sin{\left(x \right)}\, dx$$

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

   So, the result is: $6 \cos{\left(x \right)}$

5. Add the constant of integration:

   $$x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} - 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} - 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}+ \mathrm{constant}$$