What is the integral of xln(5x) ?

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

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integral of xln(5x)

\int x ln (5 x) d x

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

Solution steps

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

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

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

Add a constant to the solution

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

\frac{d}{d x} ((x^{2} - \frac{5}{4}) e^{x})

\frac{\partial}{\partial x} (sin (x^{2} \cdot y^{2}))

x^{2} \frac{d y}{d x} + 2 x y = ln (x)

\int \frac{3 x}{x ^{2} - 2} d x

\int (5 x^{2} + 3 x^{- 2}) d x

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