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Calculus · Definite-Integrals

Integrate y·(1 - x) from 0 to x

\int_{0}^{x} y \left(1 - x\right)\, dy

Step by step

  1. \int_{0}^{x} y \left(1 - x\right)\, dy

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

  2. \int y \left(1 - x\right)\, dy = 1 - x \int y\, dy

    Pull the constant 1 - x out of the integral.

  3. \int y\, dy = \frac{y^{2}}{2}

    Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

  4. F(x) - F(0) = \left(\frac{x^{2} \left(1 - x\right)}{2}\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  5. = \frac{x^{2} \left(1 - x\right)}{2}

    Simplify.

Reveal the answer
\frac{x^{2} \left(1 - x\right)}{2}