Solve ∫ (frac{sqrt{x}}{5}+4/sqrt[4]{x}) wrt x

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

Evaluate

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$\frac{2x^{\frac{3}{2}}}{15}+4\int \frac{1}{\sqrt[4]{x}}\mathrm{d}x$$

Rewrite \frac{1}{\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{3}{4}}}{\frac{3}{4}}. Simplify. Multiply 4 times \frac{4x^{\frac{3}{4}}}{3}.

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}$$

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.

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}+С$$

Differentiate w.r.t. x

$$\frac{\sqrt{x}}{5}+\frac{4}{\sqrt[4]{x}}$$