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

The integral of xln(sqrt(5x)) is 1/2 (1/2 x^2ln(5x)-(x^2)/4)+C

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integral of xln(sqrt(5x))

\int x ln (5 x) d x

\frac{1}{2} (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}) + C

Solution steps

\int x ln (5 x) d x

Simplify

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= \frac{1}{2} (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4})

Add a constant to the solution

= \frac{1}{2} (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}) + C

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What is the integral of xln(sqrt(5x)) ?
The integral of xln(sqrt(5x)) is 1/2 (1/2 x^2ln(5x)-(x^2)/4)+C