Solve any maths problem

Equations, derivatives, integrals, matrices, triangles, primes, statistics, or a word problem the tutor breaks into parts.

Calculus · Definite-Integrals

Integrate x·(pi - x)·sin(x) from 0 to pi

\int_{0}^{\pi} x \left(\pi - x\right) \sin{\left(x \right)}\, dx

Step by step

  1. \int_{0}^{\pi} x \left(\pi - x\right) \sin{\left(x \right)}\, dx

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

  2. u = - x,\quad du = -1\, dx

    Substitute u = - x.

  3. \int x \left(\pi - x\right) \sin{\left(x \right)}\, dx = \int - u^{2} \sin{\left(u \right)} - u \pi \sin{\left(u \right)}\, d_u

    Rewrite the integral in terms of u.

  4. \int - u^{2} \sin{\left(u \right)} - u \pi \sin{\left(u \right)}\, d_u = \int - u^{2} \sin{\left(u \right)}\, d_u + \int - u \pi \sin{\left(u \right)}\, d_u

    The integral of a sum is the sum of the integrals.

  5. \int - u^{2} \sin{\left(u \right)}\, d_u = -1 \int u^{2} \sin{\left(u \right)}\, d_u

    Pull the constant -1 out of the integral.

  6. u = u^{2},\quad dv = \sin{\left(u \right)}\, d_u

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

  7. du = 2 u\, d_u,\quad v = - \cos{\left(u \right)}

    Differentiate u, integrate dv.

  8. \int u^{2} \sin{\left(u \right)}\, d_u = - u^{2} \cos{\left(u \right)} - \int - 2 u \cos{\left(u \right)}\, d_u

    Apply the formula.

  9. u = - 2 u,\quad dv = \cos{\left(u \right)}\, d_u

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

  10. du = -2\, d_u,\quad v = \sin{\left(u \right)}

    Differentiate u, integrate dv.

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

    Apply the formula.

  12. \int - 2 \sin{\left(u \right)}\, d_u = -2 \int \sin{\left(u \right)}\, d_u

    Pull the constant -2 out of the integral.

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

    Standard trigonometric antiderivative.

  14. = - 2 u \sin{\left(u \right)} - 2 \cos{\left(u \right)}

    Combine.

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

    Combine.

  16. \int - u \pi \sin{\left(u \right)}\, d_u = - \pi \int u \sin{\left(u \right)}\, d_u

    Pull the constant - \pi out of the integral.

  17. u = u,\quad dv = \sin{\left(u \right)}\, d_u

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

  18. \vdots

    …

  19. = x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} - \pi \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right) - 2 \cos{\left(x \right)}

    Substitute back u = - x.

  20. F(\pi) - F(0) = \left(2\right) - \left(-2\right)

    Fundamental theorem of calculus: plug in the limits.

  21. = 4

    Simplify.

Reveal the answer
4