Solve ∫ 1/sqrt{5x wrt x

$$\int{ \frac{ 1 }{ \sqrt{ 5x } } }d x$$

Evaluate

$$\frac{2\sqrt{5x}}{5}+С$$

Solution Steps

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

$$\frac{\sqrt{5}\int \frac{1}{\sqrt{x}}\mathrm{d}x}{5}$$

Rewrite \frac{1}{\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{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form.

$$\frac{2\sqrt{5}\sqrt{x}}{5}$$

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.

$$\frac{2\sqrt{5}\sqrt{x}}{5}+С$$

Differentiate w.r.t. x

$$\frac{\sqrt{\frac{5}{x}}}{5}$$