Solve ∫ (5x^2+2/x^4-3sqrt[3]{x}) wrt x

$$\int{ (5 { x }^{ 2 } + \frac{ 2 }{ { x }^{ 4 } } -3 \sqrt[ 3 ]{ x } ) }d x$$

Evaluate

$$\frac{5x^{3}}{3}-\frac{9x^{\frac{4}{3}}}{4}-\frac{2}{3x^{3}}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 5x^{2}\mathrm{d}x+\int \frac{2}{x^{4}}\mathrm{d}x+\int -3\sqrt[3]{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{2}\mathrm{d}x+2\int \frac{1}{x^{4}}\mathrm{d}x-3\int \sqrt[3]{x}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 5 times \frac{x^{3}}{3}.

$$\frac{5x^{3}}{3}+2\int \frac{1}{x^{4}}\mathrm{d}x-3\int \sqrt[3]{x}\mathrm{d}x$$

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 2 times -\frac{1}{3x^{3}}.

$$\frac{5x^{3}}{3}-\frac{\frac{2}{x^{3}}}{3}-3\int \sqrt[3]{x}\mathrm{d}x$$

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

$$\frac{5x^{3}}{3}-\frac{\frac{2}{x^{3}}}{3}-\frac{9x^{\frac{4}{3}}}{4}$$

Simplify.

$$\frac{5x^{3}}{3}-\frac{2}{3x^{3}}-\frac{9x^{\frac{4}{3}}}{4}$$

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.

$$\frac{5x^{3}}{3}-\frac{2}{3x^{3}}-\frac{9x^{\frac{4}{3}}}{4}+С$$

Differentiate w.r.t. x

$$5x^{2}-3\sqrt[3]{x}+\frac{2}{x^{4}}$$