Solve ∫ (2/x^3-1/x^4) wrt x

$$\int ( \frac { 2 } { x ^ { 3 } } - \frac { 1 } { x ^ { 4 } } ) d x$$

Evaluate

$$\frac{1-3x}{3x^{3}}+С$$

Solution Steps

Integrate the sum term by term.

$$\int \frac{2}{x^{3}}\mathrm{d}x+\int -\frac{1}{x^{4}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int \frac{1}{x^{3}}\mathrm{d}x-\int \frac{1}{x^{4}}\mathrm{d}x$$

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}}. Multiply 2 times -\frac{1}{2x^{2}}.

$$-\frac{1}{x^{2}}-\int \frac{1}{x^{4}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{4}}\mathrm{d}x with -\frac{1}{3x^{3}}. Multiply -1 times -\frac{1}{3x^{3}}.

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

Simplify.

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

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.

$$\frac{\frac{1}{3}-x}{x^{3}}+С$$

Differentiate w.r.t. x

$$\frac{2x-1}{x^{4}}$$