Solve ∫ (from 3 to 4) of (5x^2+5) wrt x

$$\int _ { 3 } ^ { 4 } ( 5 x ^ { 2 } + 5 ) d x$$

Evaluate

$$\frac{200}{3}\approx 66.666666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$\frac{5}{3}\times 4^{3}+5\times 4-\left(\frac{5}{3}\times 3^{3}+5\times 3\right)$$

Simplify.

$$\frac{200}{3}$$