Solve ∫ (from 1 to 5) of (x^3-1/x) wrt x

$$\int _ { 1 } ^ { 5 } ( x ^ { 3 } - \frac { 1 } { x } ) d x$$

Evaluate

$$156-\ln(5)\approx 154.390562088$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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.

$$\frac{5^{4}}{4}-\ln(|5|)-\left(\frac{1^{4}}{4}-\ln(|1|)\right)$$

Simplify.

$$156-\ln(5)$$