Solve ∫ (sqrt[7]{x}-1/x^3-1/x^4+1/sqrt{x}) wrt x

$$\int ( \sqrt[ 7 ] { x } - \frac { 1 } { x ^ { 3 } } - \frac { 1 } { x ^ { 4 } } + \frac { 1 } { \sqrt { x } } ) d x$$

Evaluate

$$\frac{7x^{\frac{8}{7}}}{8}+2\sqrt{x}+\frac{1}{2x^{2}}+\frac{1}{3x^{3}}+С$$

Solution Steps

Integrate the sum term by term.

$$\int \sqrt[7]{x}\mathrm{d}x+\int -\frac{1}{x^{3}}\mathrm{d}x+\int -\frac{1}{x^{4}}\mathrm{d}x+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sqrt[7]{x}\mathrm{d}x-\int \frac{1}{x^{3}}\mathrm{d}x-\int \frac{1}{x^{4}}\mathrm{d}x+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Rewrite \sqrt[7]{x} as x^{\frac{1}{7}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{7}}\mathrm{d}x with \frac{x^{\frac{8}{7}}}{\frac{8}{7}}. Simplify.

$$\frac{7x^{\frac{8}{7}}}{8}-\int \frac{1}{x^{3}}\mathrm{d}x-\int \frac{1}{x^{4}}\mathrm{d}x+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

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

$$\frac{7x^{\frac{8}{7}}}{8}+\frac{1}{2x^{2}}-\int \frac{1}{x^{4}}\mathrm{d}x+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

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}}.

$$\frac{7x^{\frac{8}{7}}}{8}+\frac{1}{2x^{2}}+\frac{1}{3x^{3}}+\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

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.

$$\frac{7x^{\frac{8}{7}}}{8}+\frac{1}{2x^{2}}+\frac{1}{3x^{3}}+2\sqrt{x}$$

Simplify.

$$\frac{7x^{\frac{8}{7}}}{8}+2\sqrt{x}+\frac{\frac{1}{3}+\frac{x}{2}}{x^{3}}$$

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.

$$\frac{7x^{\frac{8}{7}}}{8}+2\sqrt{x}+\frac{\frac{1}{3}+\frac{x}{2}}{x^{3}}+С$$

Differentiate w.r.t. x

$$\frac{x^{\frac{29}{7}}+x^{\frac{7}{2}}-x-1}{x^{4}}$$