What is the integral of 4xln(6x) ?

The integral of 4xln(6x) is 2x^2ln(6x)-x^2+C

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integral of 4xln(6x)

\int 4 x ln (6 x) d x

2 x^{2} ln (6 x) - x^{2} + C

Solution steps

\int 4 x ln (6 x) d x

Take the constant out:

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

= 4 \cdot \int x ln (6 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} ln (6 x) - \int \frac{x}{2} d x

= 4 (\frac{1}{2} x^{2} ln (6 x) - \int \frac{x}{2} d x)

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

= 4 (\frac{1}{2} x^{2} ln (6 x) - \frac{x ^{2}}{4})

Simplify

4 (\frac{1}{2} x^{2} ln (6 x) - \frac{x ^{2}}{4}) : 2 x^{2} ln (6 x) - x^{2}

= 2 x^{2} ln (6 x) - x^{2}

Add a constant to the solution

= 2 x^{2} ln (6 x) - x^{2} + C

100 y^{^{''}} - 20 y^{^{'}} + y = 0

(x^{2} y + 3 y) y^{^{'}} = 2 x y^{2} + 8 x

\frac{d y}{d x} = e^{- 3 y}

\frac{d x}{d t} = t^{3} (1 - x), x (0) = 3

\frac{d}{d x} (3 x^{k - 2} + 2 k x^{k - 1} + 4 x^{k})