Solve any maths problem

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1. \int_{0}^{1} \frac{y \left(1 - y\right)}{2}\, dy

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

2. \int \frac{y \left(1 - y\right)}{2}\, dy = \frac{1}{2} \int y \left(1 - y\right)\, dy

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

3. u = - y,\quad du = -1\, dy

   Substitute u = - y.

4. \int y \left(1 - y\right)\, dy = \int u^{2} + u\, d_u

   Rewrite the integral in terms of u.

5. \int u^{2} + u\, d_u = \int u^{2}\, d_u + \int u\, d_u

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

6. \int u^{2}\, d_u = \frac{u^{3}}{3}

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

7. \int u\, d_u = \frac{u^{2}}{2}

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

8. = - \frac{y^{3}}{3} + \frac{y^{2}}{2}

   Substitute back u = - y.

9. F(1) - F(0) = \left(\frac{1}{12}\right) - \left(0\right)

   Fundamental theorem of calculus: plug in the limits.

10. = \frac{1}{12}

    Simplify.