Solve any maths problem

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1. \int_{- \pi}^{\pi} x^{2} \cos{\left(2 x \right)}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. u = x^{2},\quad dv = \cos{\left(2 x \right)}\, dx

   Integration by parts: ∫u dv = uv − ∫v du.

3. du = 2 x\, dx,\quad v = \frac{\sin{\left(2 x \right)}}{2}

   Differentiate u, integrate dv.

4. \int x^{2} \cos{\left(2 x \right)}\, dx = \frac{x^{2} \sin{\left(2 x \right)}}{2} - \int x \sin{\left(2 x \right)}\, dx

   Apply the formula.

5. u = x,\quad dv = \sin{\left(2 x \right)}\, dx

   Integration by parts: ∫u dv = uv − ∫v du.

6. du = 1\, dx,\quad v = - \frac{\cos{\left(2 x \right)}}{2}

   Differentiate u, integrate dv.

7. \int x \sin{\left(2 x \right)}\, dx = - \frac{x \cos{\left(2 x \right)}}{2} - \int - \frac{\cos{\left(2 x \right)}}{2}\, dx

   Apply the formula.

8. \int - \frac{\cos{\left(2 x \right)}}{2}\, dx = - \frac{1}{2} \int \cos{\left(2 x \right)}\, dx

   Pull the constant - \frac{1}{2} out of the integral.

9. u = 2 x,\quad du = 2\, dx

   Substitute u = 2 x.

10. \int \cos{\left(2 x \right)}\, dx = \int \frac{\cos{\left(u \right)}}{2}\, d_u

    Rewrite the integral in terms of u.

11. \int \frac{\cos{\left(u \right)}}{2}\, d_u = \frac{1}{2} \int \cos{\left(u \right)}\, d_u

    Pull the constant \frac{1}{2} out of the integral.

12. \int \cos{\left(u \right)}\, d_u = \sin{\left(u \right)}

    Standard trigonometric antiderivative.

13. = \frac{\sin{\left(2 x \right)}}{2}

    Substitute back u = 2 x.

14. = - \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}

    Combine.

15. = \frac{x^{2} \sin{\left(2 x \right)}}{2} + \frac{x \cos{\left(2 x \right)}}{2} - \frac{\sin{\left(2 x \right)}}{4}

    Combine.

16. F(\pi) - F(- \pi) = \left(\frac{\pi}{2}\right) - \left(- \frac{\pi}{2}\right)

    Fundamental theorem of calculus: plug in the limits.

17. = \pi \approx 3.1416

    Simplify.