Solve ∫ (1/sqrt{x}-1/sqrt[3]{x^2}-1/sqrt[4]{x^3}) wrt x=

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

Evaluate

$$2\sqrt{x}-3\sqrt[3]{x}-4\sqrt[4]{x}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Rewrite \frac{1}{\sqrt{x}} as x^{-\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form.

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

Rewrite \frac{1}{x^{\frac{2}{3}}} as x^{-\frac{2}{3}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{2}{3}}\mathrm{d}x with \frac{x^{\frac{1}{3}}}{\frac{1}{3}}. Simplify and convert from exponential to radical form. Multiply -1 times 3\sqrt[3]{x}.

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

Rewrite \frac{1}{x^{\frac{3}{4}}} as x^{-\frac{3}{4}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{3}{4}}\mathrm{d}x with \frac{x^{\frac{1}{4}}}{\frac{1}{4}}. Simplify and convert from exponential to radical form. Multiply -1 times 4\sqrt[4]{x}.

$$2\sqrt{x}-3\sqrt[3]{x}-4\sqrt[4]{x}$$

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.

$$2\sqrt{x}-3\sqrt[3]{x}-4\sqrt[4]{x}+С$$

Differentiate w.r.t. x

$$\frac{1}{\sqrt{x}}-\frac{1}{x^{\frac{2}{3}}}-\frac{1}{x^{\frac{3}{4}}}$$