Integral of t²costdt dx

1. Use integration by parts:

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

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

   Then $\operatorname{du}{\left(t \right)} = 2 t$.

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

   1. The integral of cosine is sine:

      $$\int \cos{\left(t \right)}\, dt = \sin{\left(t \right)}$$

   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 $u{\left(t \right)} = 2 t$ and let $\operatorname{dv}{\left(t \right)} = \sin{\left(t \right)}$.

   Then $\operatorname{du}{\left(t \right)} = 2$.

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

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(t \right)}\, dt = - \cos{\left(t \right)}$$

   Now evaluate the sub-integral.

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

   $$\int \left(- 2 \cos{\left(t \right)}\right)\, dt = - 2 \int \cos{\left(t \right)}\, dt$$

   1. The integral of cosine is sine:

      $$\int \cos{\left(t \right)}\, dt = \sin{\left(t \right)}$$

   So, the result is: $- 2 \sin{\left(t \right)}$

4. Add the constant of integration:

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

The answer is: