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

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

Evaluate

$$\frac{65}{2}=32.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{2}+5x+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int 5x\mathrm{d}x+\int 1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x+5\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$x^{3}+5\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$x^{3}+\frac{5x^{2}}{2}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}+\frac{5x^{2}}{2}+x$$

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.

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

Simplify.

$$\frac{65}{2}$$