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

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

Evaluate

$$\frac{80}{3}\approx 26.666666667$$

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.

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

Simplify.

$$\frac{80}{3}$$