Solve ∫ (2+sqrt[4]{x})

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

Evaluate

$$\frac{4x^{\frac{5}{4}}}{5}+2x+С$$

Solution Steps

Integrate the sum term by term.

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

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$2x+\frac{4x^{\frac{5}{4}}}{5}$$

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.

$$2x+\frac{4x^{\frac{5}{4}}}{5}+С$$

Differentiate w.r.t. x

$$\sqrt[4]{x}+2$$