Solve ∫ (from 1 to 3) of (6x^2-4x+5) wrt x

$$\int _ { 1 } ^ { 3 } ( 6 x ^ { 2 } - 4 x + 5 ) d x$$

Evaluate

$$46$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6x^{2}-4x+5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 6x^{2}\mathrm{d}x+\int -4x\mathrm{d}x+\int 5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\int x^{2}\mathrm{d}x-4\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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

$$2x^{3}-4\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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}.

$$2x^{3}-2x^{2}+\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x^{3}-2x^{2}+5x$$

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.

$$2\times 3^{3}-2\times 3^{2}+5\times 3-\left(2\times 1^{3}-2\times 1^{2}+5\times 1\right)$$

Simplify.

$$46$$