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

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

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

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

2 x^{2} ln (x - 1) - 2 x ln (x - 1) - x^{2} + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

= 2 x^{2} ln (x - 1) - 2 x ln (x - 1) - x^{2}

Add a constant to the solution

= 2 x^{2} ln (x - 1) - 2 x ln (x - 1) - x^{2} + C

\int (5 x^{3} + 10 x)^{3} (3 x^{2} + 2) d x

x^{5} + 5^{x}

\int_{0}^{1} sin (x) d x

x \to 0 lim (\frac{2 x ^{2}}{sin ^{2} ( 8 x )})

f (x) = x^{4} - 2 x^{3} - 5 x - 3, at x = 1

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