Solve ∫ 1/2frac{sqrt{5}}{sqrt[4]{x wrt x?

$$\int \frac { 1 } { 2 } \frac { \sqrt { 5 } } { \sqrt[ 4 ] { x } } d x ?$$

Evaluate

$$\frac{2\sqrt{5}x^{\frac{3}{4}}}{3}+С$$

Solution Steps

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

$$\frac{\sqrt{5}}{2}\int \frac{1}{\sqrt[4]{x}}\mathrm{d}x$$

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.

$$\frac{\sqrt{5}}{2}\times \frac{4x^{\frac{3}{4}}}{3}$$

Simplify.

$$\frac{2\sqrt{5}x^{\frac{3}{4}}}{3}$$

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.

$$\frac{2\sqrt{5}x^{\frac{3}{4}}}{3}+С$$

Differentiate w.r.t. x

$$\frac{\sqrt{5}}{2\sqrt[4]{x}}$$