What is the integral of ln(1-4x) ?

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

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integral of ln(1-4x)

\int ln (1 - 4 x) d x

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

Solution steps

\int ln (1 - 4 x) d x

Apply u-substitution

= \int - \frac{1}{4} ln (u) d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int ln (u) d u

Apply Integration By Parts

= - \frac{1}{4} (u ln (u) - \int 1 d u)

\int 1 d u = u

= - \frac{1}{4} (u ln (u) - u)

Substitute back

u = 1 - 4 x

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

Simplify

- \frac{1}{4} ((1 - 4 x) ln (1 - 4 x) - (1 - 4 x)) : - \frac{1}{4} ln (1 - 4 x) + x ln (1 - 4 x) + \frac{1}{4} - x

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

Add a constant to the solution

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

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