Solve ∫ (from 0 to 3) of (-3x^2+12x+2)(-x) wrt x

$$\int_{ 0 }^{ 3 } (-3 { x }^{ 2 } +12x+2)(-x) d x$$

Evaluate

$$-\frac{225}{4}=-56.25$$

Solution Steps

Use the distributive property to multiply -3x^{2}+12x+2 by -x.

$$\int _{0}^{3}-3x^{2}\left(-x\right)+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

Multiply -3 and -1 to get 3.

$$\int _{0}^{3}3x^{2}x+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int _{0}^{3}3x^{3}+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

Multiply x and x to get x^{2}.

$$\int _{0}^{3}3x^{3}+12x^{2}\left(-1\right)+2\left(-1\right)x\mathrm{d}x$$

Multiply 12 and -1 to get -12.

$$\int _{0}^{3}3x^{3}-12x^{2}+2\left(-1\right)x\mathrm{d}x$$

Multiply 2 and -1 to get -2.

$$\int _{0}^{3}3x^{3}-12x^{2}-2x\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}. Multiply 3 times \frac{x^{4}}{4}.

$$\frac{3x^{4}}{4}-12\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x$$

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 -12 times \frac{x^{3}}{3}.

$$\frac{3x^{4}}{4}-4x^{3}-2\int x\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}-4x^{3}-x^{2}$$

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.

$$\frac{3}{4}\times 3^{4}-4\times 3^{3}-3^{2}-\left(\frac{3}{4}\times 0^{4}-4\times 0^{3}-0^{2}\right)$$

Simplify.

$$-\frac{225}{4}$$