Solve ∫ (from -2 to -1) of 1/x^2-x^2 wrt x

$$\int_{ -2 }^{ -1 } \frac{ 1 }{ { x }^{ 2 } } - { x }^{ 2 } d x$$

Evaluate

$$-\frac{11}{6}\approx -1.833333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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 -1 times \frac{x^{3}}{3}.

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

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.

$$-\left(-1\right)^{-1}-\frac{\left(-1\right)^{3}}{3}-\left(-\left(-2\right)^{-1}-\frac{\left(-2\right)^{3}}{3}\right)$$

Simplify.

$$-\frac{11}{6}$$