Solve ∫ (from -2 to 2) of (pi)-x^2+4 wrt x

$$\int_{ -2 }^{ 2 } ( \pi )- { x }^{ 2 } +4 d x$$

Evaluate

$$4\pi +\frac{32}{3}\approx 23.233037281$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \pi -x^{2}+4\mathrm{d}x$$

Integrate the sum term by term.

$$\int \pi \mathrm{d}x+\int -x^{2}\mathrm{d}x+\int 4\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \pi \mathrm{d}x-\int x^{2}\mathrm{d}x+\int 4\mathrm{d}x$$

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

$$\pi x-\int x^{2}\mathrm{d}x+\int 4\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

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

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

$$4\pi +\frac{32}{3}$$