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

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

Evaluate

$$1$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

$$\frac{x^{2}}{2}+\frac{1}{x}$$

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.

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

Simplify.

$$1$$