Solve ∫ (from 1 to 3) of (3x^2-1/x^2)

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

Evaluate

$$\frac{76}{3}\approx 25.333333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{2}}\mathrm{d}x with -\frac{1}{x}. Multiply -1 times -\frac{1}{x}.

$$x^{3}+\frac{1}{x}$$

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.

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

Simplify.

$$\frac{76}{3}$$