Solve ∫ (from 0 to 3) of (12x^11-90x^9+4x+2) wrt x

$$\int _ { 0 } ^ { 3 } ( 12 x ^ { 11 } - 90 x ^ { 9 } + 4 x + 2 ) d x$$

Evaluate

$$24$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 12x^{11}-90x^{9}+4x+2\mathrm{d}x$$

Integrate the sum term by term.

$$\int 12x^{11}\mathrm{d}x+\int -90x^{9}\mathrm{d}x+\int 4x\mathrm{d}x+\int 2\mathrm{d}x$$

Factor out the constant in each of the terms.

$$12\int x^{11}\mathrm{d}x-90\int x^{9}\mathrm{d}x+4\int x\mathrm{d}x+\int 2\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{11}\mathrm{d}x with \frac{x^{12}}{12}. Multiply 12 times \frac{x^{12}}{12}.

$$x^{12}-90\int x^{9}\mathrm{d}x+4\int x\mathrm{d}x+\int 2\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{9}\mathrm{d}x with \frac{x^{10}}{10}. Multiply -90 times \frac{x^{10}}{10}.

$$x^{12}-9x^{10}+4\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

$$x^{12}-9x^{10}+2x^{2}+\int 2\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{12}-9x^{10}+2x^{2}+2x$$

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.

$$3^{12}-9\times 3^{10}+2\times 3^{2}+2\times 3-\left(0^{12}-9\times 0^{10}+2\times 0^{2}+2\times 0\right)$$

Simplify.

$$24$$