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

    This one needs a special technique; the computer algebra system gives the antiderivative directly.

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

    Fundamental theorem of calculus: plug in the limits.

  4. = 4

    Simplify.

Reveal the answer
4