Solve ∫ (from 2 to 5) of x^2+3 wrt x

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

Evaluate

$$48$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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

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

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.

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

Simplify.

$$48$$