Solve ∫ (from a to frac { 1) of {2}}(x-1)*x wrt x

$$\int _ { a } ^ { \frac { 1 } { 2 } } ( x - 1 ) \times x d x$$

Evaluate

$$-\frac{a^{3}}{3}+\frac{a^{2}}{2}-\frac{1}{12}$$

Solution Steps

Use the distributive property to multiply x-1 by x.

$$\int _{a}^{\frac{1}{2}}x^{2}-x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\frac{x^{3}}{3}-\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -1 times \frac{x^{2}}{2}.

$$\frac{x^{3}}{3}-\frac{x^{2}}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{\left(\frac{1}{2}\right)^{3}}{3}-\frac{\left(\frac{1}{2}\right)^{2}}{2}-\left(\frac{a^{3}}{3}-\frac{a^{2}}{2}\right)$$

Simplify.

$$-\frac{1}{12}-\frac{a^{3}}{3}+\frac{a^{2}}{2}$$

Differentiate w.r.t. a

$$a\left(1-a\right)$$