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Calculus · Definite-Integrals

Integrate 2x·(1 - x)·sin(pi·x) from 0 to 1

\int_{0}^{1} 2 x \left(1 - x\right) \sin{\left(\pi x \right)}\, dx

Step by step

  1. \int_{0}^{1} 2 x \left(1 - x\right) \sin{\left(\pi x \right)}\, dx

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

  2. \int 2 x \left(1 - x\right) \sin{\left(\pi x \right)}\, dx = 2 \int x \left(1 - x\right) \sin{\left(\pi x \right)}\, dx

    Pull the constant 2 out of the integral.

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

    Substitute u = - x.

  4. \int x \left(1 - x\right) \sin{\left(\pi x \right)}\, dx = \int - u^{2} \sin{\left(u \pi \right)} - u \sin{\left(u \pi \right)}\, d_u

    Rewrite the integral in terms of u.

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

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

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

    Pull the constant -1 out of the integral.

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

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

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

    Differentiate u, integrate dv.

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

    Apply the formula.

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

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

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

    Differentiate u, integrate dv.

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

    Apply the formula.

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

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

  14. u = u \pi,\quad du = \pi\, d_u

    Substitute u = u \pi.

  15. \int \sin{\left(u \pi \right)}\, d_u = \int \frac{\sin{\left(u \right)}}{\pi}\, d_u

    Rewrite the integral in terms of u.

  16. \int \frac{\sin{\left(u \right)}}{\pi}\, d_u = \frac{1}{\pi} \int \sin{\left(u \right)}\, d_u

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

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

    Standard trigonometric antiderivative.

  18. \vdots

    …

  19. = \frac{x^{2} \cos{\left(\pi x \right)}}{\pi} - \frac{2 x \sin{\left(\pi x \right)}}{\pi^{2}} - \frac{x \cos{\left(\pi x \right)}}{\pi} + \frac{\sin{\left(\pi x \right)}}{\pi^{2}} - \frac{2 \cos{\left(\pi x \right)}}{\pi^{3}}

    Substitute back u = - x.

  20. F(1) - F(0) = \left(\frac{4}{\pi^{3}}\right) - \left(- \frac{4}{\pi^{3}}\right)

    Fundamental theorem of calculus: plug in the limits.

  21. = \frac{8}{\pi^{3}} \approx 0.25801

    Simplify.

Reveal the answer
\frac{8}{\pi^{3}}