Solve ∫ (from 0 to 3) of 5x^2-13x wrt x

$$\int _ { 0 } ^ { 3 } 5 x ^ { 2 } - 13 x d x$$

Evaluate

$$-\frac{27}{2}=-13.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5x^{2}-13x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x^{2}\mathrm{d}x+\int -13x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{2}\mathrm{d}x-13\int x\mathrm{d}x$$

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}. Multiply 5 times \frac{x^{3}}{3}.

$$\frac{5x^{3}}{3}-13\int x\mathrm{d}x$$

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 -13 times \frac{x^{2}}{2}.

$$\frac{5x^{3}}{3}-\frac{13x^{2}}{2}$$

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.

$$\frac{5}{3}\times 3^{3}-\frac{13}{2}\times 3^{2}-\left(\frac{5}{3}\times 0^{3}-\frac{13}{2}\times 0^{2}\right)$$

Simplify.

$$-\frac{27}{2}$$