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

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

Evaluate

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

Solution Steps

Integrate the sum term by term.

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

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{3x^{\frac{5}{3}}}{5}.

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

Rewrite \sqrt[4]{x} as x^{\frac{1}{4}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{4}}\mathrm{d}x with \frac{x^{\frac{5}{4}}}{\frac{5}{4}}. Simplify.

$$\frac{3x^{\frac{5}{3}}+4x^{\frac{5}{4}}}{5}$$

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.

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

Differentiate w.r.t. x

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