Solve ∫ (from - 5 to - 4) of (3x^2+24x+49) wrt x

$$\int _ { - 5 } ^ { - 4 } ( 3 x ^ { 2 } + 24 x + 49 ) d x$$

Evaluate

$$2$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{2}+24x+49\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int 24x\mathrm{d}x+\int 49\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x+24\int x\mathrm{d}x+\int 49\mathrm{d}x$$

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

$$x^{3}+24\int x\mathrm{d}x+\int 49\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 24 times \frac{x^{2}}{2}.

$$x^{3}+12x^{2}+\int 49\mathrm{d}x$$

Find the integral of 49 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}+12x^{2}+49x$$

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.

$$\left(-4\right)^{3}+12\left(-4\right)^{2}+49\left(-4\right)-\left(\left(-5\right)^{3}+12\left(-5\right)^{2}+49\left(-5\right)\right)$$

Simplify.

$$2$$