Questions: ∫ 1/√(2x) dx

Solution Steps

To solve the integral \(\int \frac{1}{\sqrt{2x}} \, dx\), we can use a substitution method. Let \(u = 2x\), then \(du = 2dx\). This substitution simplifies the integral into a more manageable form.

To solve the integral \(\int \frac{1}{\sqrt{2x}} \, dx\), we use the substitution \(u = 2x\). Then, \(du = 2dx\) or \(dx = \frac{du}{2}\).

Substitute \(u\) and \(dx\) into the integral: \[ \int \frac{1}{\sqrt{2x}} \, dx = \int \frac{1}{\sqrt{u}} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{1}{\sqrt{u}} \, du \]

The integral \(\int \frac{1}{\sqrt{u}} \, du\) is a standard integral: \[ \int \frac{1}{\sqrt{u}} \, du = 2\sqrt{u} \] Thus, \[ \frac{1}{2} \int \frac{1}{\sqrt{u}} \, du = \frac{1}{2} \cdot 2\sqrt{u} = \sqrt{u} \]

Replace \(u\) with \(2x\): \[ \sqrt{u} = \sqrt{2x} \]

Final Answer