What is the integral of ln(9x+7) ?

The integral of ln(9x+7) is xln(9x+7)+7/9 ln(9x+7)-x-7/9+C

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integral of ln(9x+7)

\int ln (9 x + 7) d x

x ln (9 x + 7) + \frac{7}{9} ln (9 x + 7) - x - \frac{7}{9} + C

Solution steps

\int ln (9 x + 7) 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 + 7

= \frac{1}{9} ((9 x + 7) ln (9 x + 7) - (9 x + 7))

Simplify

\frac{1}{9} ((9 x + 7) ln (9 x + 7) - (9 x + 7)) : x ln (9 x + 7) + \frac{7}{9} ln (9 x + 7) - x - \frac{7}{9}

= x ln (9 x + 7) + \frac{7}{9} ln (9 x + 7) - x - \frac{7}{9}

Add a constant to the solution

= x ln (9 x + 7) + \frac{7}{9} ln (9 x + 7) - x - \frac{7}{9} + C

f (x) = (2 x^{2} - 3)^{2}

\int_{1}^{3} (3 x^{2} + 5 x - 4) d x

s - 1

\int e^{- x} cos (x) d x

\frac{d}{d x} (a x^{2} + b)

What is the integral of ln(9x+7) ?
The integral of ln(9x+7) is xln(9x+7)+7/9 ln(9x+7)-x-7/9+C