Solve ∫ (from 0 to 5) of 2x^2+c

$$\int _ { 0 } ^ { 5 } 2 x ^ { 2 } + c$$

Evaluate

$$5c+\frac{250}{3}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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.

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

Simplify.

$$\frac{250}{3}+5c$$

Differentiate w.r.t. c

$$5$$