What is the integral of 3x(log_{e}(5x)) ?

The integral of 3x(log_{e}(5x)) is 3(1/2 x^2ln(5x)-(x^2)/4)+C

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integral of 3x(log_{e}(5x))

\int 3 x (lo g_{e} (5 x)) d x

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

Solution steps

\int 3 x lo g_{e} (5 x) d x

Take the constant out:

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

= 3 \cdot \int x lo g_{e} (5 x) d x

Simplify

lo g_{e} (5 x) : ln (5 x)

= 3 \cdot \int x ln (5 x) d x

Apply Integration By Parts

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

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

= 3 (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4})

Add a constant to the solution

= 3 (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}) + C

What is the integral of 3x(log_{e}(5x)) ?
The integral of 3x(log_{e}(5x)) is 3(1/2 x^2ln(5x)-(x^2)/4)+C