Solve ∫ (from - 2 to 2) of (1+x)x^2+1

$$\int _ { - 2 } ^ { 2 } ( 1 + x ) x ^ { 2 } + 1$$

Evaluate

$$\frac{28}{3}\approx 9.333333333$$

Solution Steps

Use the distributive property to multiply 1+x by x^{2}.

$$\int _{-2}^{2}x^{2}+x^{3}+1\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}+x^{3}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$\frac{x^{3}}{3}+\frac{x^{4}}{4}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+\frac{x^{4}}{4}+x$$

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.

$$\frac{2^{3}}{3}+\frac{2^{4}}{4}+2-\left(\frac{\left(-2\right)^{3}}{3}+\frac{\left(-2\right)^{4}}{4}-2\right)$$

Simplify.

$$\frac{28}{3}$$