Solve ∫ (from 2 to 3) of (x^2+1/x+4)

$$\int _ { 2 } ^ { 3 } ( x ^ { 2 } + \frac { 1 } { x } + 4 )$$

Evaluate

$$\ln(\frac{3}{2})+\frac{31}{3}\approx 10.738798441$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+\frac{1}{x}+4\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int \frac{1}{x}\mathrm{d}x+\int 4\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int \frac{1}{x}\mathrm{d}x+\int 4\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$\frac{x^{3}}{3}+\ln(|x|)+\int 4\mathrm{d}x$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+\ln(|x|)+4x$$

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.

$$\frac{3^{3}}{3}+\ln(|3|)+4\times 3-\left(\frac{2^{3}}{3}+\ln(|2|)+4\times 2\right)$$

Simplify.

$$\frac{31}{3}+\ln(\frac{3}{2})$$