Solve ∫ (from - 1.5 to 5) of -3x^2+11x+25 wrt x

$$\int _ { - 1.5 } ^ { 5 } - 3 x ^ { 2 } + 11 x + 25 d x$$

Evaluate

$$159.25$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -3x^{2}+11x+25\mathrm{d}x$$

Integrate the sum term by term.

$$\int -3x^{2}\mathrm{d}x+\int 11x\mathrm{d}x+\int 25\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-3\int x^{2}\mathrm{d}x+11\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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}.

$$-x^{3}+11\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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 11 times \frac{x^{2}}{2}.

$$-x^{3}+\frac{11x^{2}}{2}+\int 25\mathrm{d}x$$

Find the integral of 25 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-x^{3}+\frac{11x^{2}}{2}+25x$$

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.

$$-5^{3}+\frac{11}{2}\times 5^{2}+25\times 5-\left(-\left(-1.5\right)^{3}+\frac{11}{2}\left(-1.5\right)^{2}+25\left(-1.5\right)\right)$$

Simplify.

$$\frac{637}{4}$$