What is the integral of ln((2-3x)^4) ?

The integral of ln((2-3x)^4) is 4xln(2-3x)+12(-1/3 x-2/9 ln|-3x+2|)+C

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integral of ln((2-3x)^4)

\int ln ((2 - 3 x)^{4}) d x

4 x ln (2 - 3 x) + 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣) + C

Solution steps

\int ln ((2 - 3 x)^{4}) d x

Apply Integration By Parts

= x ln ((2 - 3 x)^{4}) - \int - \frac{12 x}{2 - 3 x} d x

\int - \frac{12 x}{2 - 3 x} d x = - 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣)

= x ln ((2 - 3 x)^{4}) - (- 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣))

Simplify

x ln ((2 - 3 x)^{4}) - (- 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣)) : 4 x ln (2 - 3 x) + 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣)

= 4 x ln (2 - 3 x) + 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣)

Add a constant to the solution

= 4 x ln (2 - 3 x) + 12 (- \frac{1}{3} x - \frac{2}{9} ln ∣ - 3 x + 2 ∣) + C

What is the integral of ln((2-3x)^4) ?
The integral of ln((2-3x)^4) is 4xln(2-3x)+12(-1/3 x-2/9 ln|-3x+2|)+C