Solve ∫ wrt x/sqrt[3]{8x^2}

$$\int \frac { d x } { \sqrt[ 3 ] { 8 x ^ { 2 } } }$$

Evaluate

$$\frac{3\sqrt[3]{x}}{2}+С$$

Solution Steps

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

$$\frac{\int \frac{1}{\sqrt[3]{x^{2}}}\mathrm{d}x}{\sqrt[3]{8}}$$

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.

$$\frac{3\sqrt[3]{x}}{\sqrt[3]{8}}$$

Simplify.

$$\frac{3\sqrt[3]{x}}{2}$$

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.

$$\frac{3\sqrt[3]{x}}{2}+С$$

Differentiate w.r.t. x

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