What is the integral of 3(2x-1)(ln(x-2)) ?

The integral of 3(2x-1)(ln(x-2)) is 3(x^2ln(x-2)-1/2 x^2-x-xln(x-2)-2ln(x-2)+4)+C

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integral of 3(2x-1)(ln(x-2))

\int 3 (2 x - 1) (ln (x - 2)) d x

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

Solution steps

\int 3 (2 x - 1) ln (x - 2) d x

Take the constant out:

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

= 3 \cdot \int (2 x - 1) ln (x - 2) d x

Expand

(2 x - 1) ln (x - 2) : 2 x ln (x - 2) - ln (x - 2)

= 3 \cdot \int 2 x ln (x - 2) - ln (x - 2) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int 2 x ln (x - 2) d x - \int ln (x - 2) d x)

\int 2 x ln (x - 2) d x = 2 (ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12))

\int ln (x - 2) d x = x ln (x - 2) - 2 ln (x - 2) - x + 2

= 3 (2 (ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12)) - (x ln (x - 2) - 2 ln (x - 2) - x + 2))

Simplify

3 (2 (ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12)) - (x ln (x - 2) - 2 ln (x - 2) - x + 2)) : 3 (x^{2} ln (x - 2) - \frac{1}{2} x^{2} - x - x ln (x - 2) - 2 ln (x - 2) + 4)

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

Add a constant to the solution

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

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

f (x) = \frac{- 5 x}{1 - x}

\frac{d}{d x} (a t^{m} + b t^{m + n})

\frac{d}{d x} (sin (x) sin (3 x))

4 x^{2} - 2 x + 4

What is the integral of 3(2x-1)(ln(x-2)) ?
The integral of 3(2x-1)(ln(x-2)) is 3(x^2ln(x-2)-1/2 x^2-x-xln(x-2)-2ln(x-2)+4)+C