Integral of x^5*sin(x) dx

1. Use integration by parts:

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

   Let $u{\left(x \right)} = x^{5}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 5 x^{4}$.

   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.

2. Use integration by parts:

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

   Let $u{\left(x \right)} = - 5 x^{4}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = - 20 x^{3}$.

   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.

3. Use integration by parts:

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

   Let $u{\left(x \right)} = - 20 x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = - 60 x^{2}$.

   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.

4. Use integration by parts:

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

   Let $u{\left(x \right)} = 60 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 120 x$.

   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.

5. Use integration by parts:

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

   Let $u{\left(x \right)} = 120 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 120$.

   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.

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

   $$\int \left(- 120 \cos{\left(x \right)}\right)\, dx = - 120 \int \cos{\left(x \right)}\, dx$$

   1. The integral of cosine is sine:

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

   So, the result is: $- 120 \sin{\left(x \right)}$

7. Add the constant of integration:

   $$- x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)} + 20 x^{3} \cos{\left(x \right)} - 60 x^{2} \sin{\left(x \right)} - 120 x \cos{\left(x \right)} + 120 \sin{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)} + 20 x^{3} \cos{\left(x \right)} - 60 x^{2} \sin{\left(x \right)} - 120 x \cos{\left(x \right)} + 120 \sin{\left(x \right)}+ \mathrm{constant}$$