Solve ∫ (from 1 to 15) of 2x^2-5x+3 wrt x

$$\int _ { 1 } ^ { 15 } 2 x ^ { 2 } - 5 x + 3 d x$$

Evaluate

$$\frac{5194}{3}\approx 1731.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x^{2}-5x+3\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int -5x\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x-5\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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 2 times \frac{x^{3}}{3}.

$$\frac{2x^{3}}{3}-5\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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 -5 times \frac{x^{2}}{2}.

$$\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+\int 3\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$\frac{2}{3}\times 15^{3}-\frac{5}{2}\times 15^{2}+3\times 15-\left(\frac{2}{3}\times 1^{3}-\frac{5}{2}\times 1^{2}+3\times 1\right)$$

Simplify.

$$\frac{5194}{3}$$