What is the integral of a/(sqrt(2x)) ?

The integral of a/(sqrt(2x)) is a(sqrt(x/2)+1/(sqrt(2))sqrt(x))+C

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integral of a/(sqrt(2x))

\int \frac{a}{2 x} d x

a (\frac{x}{2} + \frac{1}{2} x) + C

Solution steps

\int \frac{a}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a \cdot \int \frac{1}{2 x} d x

Apply Integration By Parts

= a (\frac{x}{2} - \int - \frac{1}{2 2 x ^{\frac{1}{2}}} d x)

\int - \frac{1}{2 2 x ^{\frac{1}{2}}} d x = - \frac{1}{2} x

= a (\frac{x}{2} - (- \frac{1}{2} x))

Simplify

= a (\frac{x}{2} + \frac{1}{2} x)

Add a constant to the solution

= a (\frac{x}{2} + \frac{1}{2} x) + C

What is the integral of a/(sqrt(2x)) ?
The integral of a/(sqrt(2x)) is a(sqrt(x/2)+1/(sqrt(2))sqrt(x))+C