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

The integral of sqrt(5x)ln(x/5) is sqrt(5) 2/3 x^{3/2}ln(x/5)-(4sqrt(5))/9 x^{3/2}+C

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integral of sqrt(5x)ln(x/5)

\int 5 x ln (\frac{x}{5}) d x

5 \frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{5}) - \frac{4 5}{9} x^{\frac{3}{2}} + C

Solution steps

\int 5 x ln (\frac{x}{5}) d x

Apply Integration By Parts

: 5 \frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{5}) - \int \frac{2 5}{3} x^{\frac{1}{2}} d x

= 5 \frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{5}) - \int \frac{2 5}{3} x^{\frac{1}{2}} d x

\int \frac{2 5}{3} x^{\frac{1}{2}} d x = \frac{4 5}{9} x^{\frac{3}{2}}

= 5 \frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{5}) - \frac{4 5}{9} x^{\frac{3}{2}}

Add a constant to the solution

= 5 \frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{5}) - \frac{4 5}{9} x^{\frac{3}{2}} + C

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What is the integral of sqrt(5x)ln(x/5) ?
The integral of sqrt(5x)ln(x/5) is sqrt(5) 2/3 x^{3/2}ln(x/5)-(4sqrt(5))/9 x^{3/2}+C