Solve ∫ (from - 3 to - 1) of x^2+4+7 wrt x

$$\int _ { - 3 } ^ { - 1 } x ^ { 2 } + 4 + 7 d x$$

Evaluate

$$\frac{92}{3}\approx 30.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+4+7\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 4\mathrm{d}x+\int 7\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int 4\mathrm{d}x+\int 7\mathrm{d}x$$

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

$$\frac{x^{3}}{3}+4x+\int 7\mathrm{d}x$$

Find the integral of 7 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+4x+7x$$

Simplify.

$$\frac{x^{3}}{3}+11x$$

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.

$$\frac{\left(-1\right)^{3}}{3}+11\left(-1\right)-\left(\frac{\left(-3\right)^{3}}{3}+11\left(-3\right)\right)$$

Simplify.

$$\frac{92}{3}$$