Solve ∫ sqrt[3]{x^1}a wrt x

$$\int{ \sqrt[ 3 ]{ { x }^{ 1 } } a }d x$$

Evaluate

$$\frac{3ax^{\frac{4}{3}}}{4}+С$$

Solution Steps

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

$$a\int \sqrt[3]{x}\mathrm{d}x$$

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.

$$a\times \frac{3x^{\frac{4}{3}}}{4}$$

Simplify.

$$\frac{3ax^{\frac{4}{3}}}{4}$$

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.

$$\frac{3ax^{\frac{4}{3}}}{4}+С$$

Differentiate w.r.t. x

$$\sqrt[3]{x}a$$