What is the integral of ln(5sqrt(x)) ?

The integral of ln(5sqrt(x)) is x(ln(5)+1/2 ln(x))-1/2 x+C

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integral of ln(5sqrt(x))

\int ln (5 x) d x

x (ln (5) + \frac{1}{2} ln (x)) - \frac{1}{2} x + C

Solution steps

\int ln (5 x) d x

Apply Integration By Parts

: x ln (5 x) - \int \frac{1}{2} d x

= x ln (5 x) - \int \frac{1}{2} d x

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

= x ln (5 x) - \frac{1}{2} x

Simplify

ln (5 x) : \frac{1}{2} ln (x) + ln (5)

= x (ln (5) + \frac{1}{2} ln (x)) - \frac{1}{2} x

Add a constant to the solution

= x (ln (5) + \frac{1}{2} ln (x)) - \frac{1}{2} x + C

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What is the integral of ln(5sqrt(x)) ?
The integral of ln(5sqrt(x)) is x(ln(5)+1/2 ln(x))-1/2 x+C