What is the integral of ln(9x+1) ?

The integral of ln(9x+1) is xln(9x+1)+1/9 ln(9x+1)-x-1/9+C

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integral of ln(9x+1)

\int ln (9 x + 1) d x

x ln (9 x + 1) + \frac{1}{9} ln (9 x + 1) - x - \frac{1}{9} + C

Solution steps

\int ln (9 x + 1) d x

Apply u-substitution

: \int ln (u) \frac{1}{9} d u

= \int ln (u) \frac{1}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{9} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{9} (u ln (u) - u)

Substitute back

u = 9 x + 1

= \frac{1}{9} ((9 x + 1) ln (9 x + 1) - (9 x + 1))

Simplify

\frac{1}{9} ((9 x + 1) ln (9 x + 1) - (9 x + 1)) : x ln (9 x + 1) + \frac{1}{9} ln (9 x + 1) - x - \frac{1}{9}

= x ln (9 x + 1) + \frac{1}{9} ln (9 x + 1) - x - \frac{1}{9}

Add a constant to the solution

= x ln (9 x + 1) + \frac{1}{9} ln (9 x + 1) - x - \frac{1}{9} + C

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\int_{0}^{2 π} sin^{2} (t) d t

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What is the integral of ln(9x+1) ?
The integral of ln(9x+1) is xln(9x+1)+1/9 ln(9x+1)-x-1/9+C