Solve ∫ 1/x^231/x wrt x

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

Evaluate

$$-\frac{3}{2x^{2}}+С$$

Solution Steps

Express \frac{1}{x^{2}}\times 3 as a single fraction.

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

Multiply \frac{3}{x^{2}} times \frac{1}{x} by multiplying numerator times numerator and denominator times denominator.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{3}}\mathrm{d}x with -\frac{1}{2x^{2}}.

$$-\frac{\frac{3}{x^{2}}}{2}$$

Simplify.

$$-\frac{3}{2x^{2}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-\frac{3}{2x^{2}}+С$$

Differentiate w.r.t. x

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