Solve ∫ (from 1 to 3) of (x^2-1)x wrt x

$$\int _ { 1 } ^ { 3 } ( x ^ { 2 } - 1 ) x d x$$

Evaluate

$$16$$

Solution Steps

Use the distributive property to multiply x^{2}-1 by x.

$$\int _{1}^{3}x^{3}-x\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$\frac{x^{4}}{4}-\int x\mathrm{d}x$$

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}.

$$\frac{x^{4}}{4}-\frac{x^{2}}{2}$$

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.

$$\frac{3^{4}}{4}-\frac{3^{2}}{2}-\left(\frac{1^{4}}{4}-\frac{1^{2}}{2}\right)$$

Simplify.

$$16$$