Solve ∫ (from 5 to 10) of 2x^2+7x wrt x

$$\int _ { 5 } ^ { 10 } 2 x ^ { 2 } + 7 x d x$$

Evaluate

$$\frac{5075}{6}\approx 845.833333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x^{2}+7x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int 7x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x+7\int x\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}+7\int x\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}+\frac{7x^{2}}{2}$$

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.

$$\frac{2}{3}\times 10^{3}+\frac{7}{2}\times 10^{2}-\left(\frac{2}{3}\times 5^{3}+\frac{7}{2}\times 5^{2}\right)$$

Simplify.

$$\frac{5075}{6}$$