What is the integral of ln(6x+7) ?

The integral of ln(6x+7) is xln(6x+7)+7/6 ln(6x+7)-x-7/6+C

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integral of ln(6x+7)

\int ln (6 x + 7) d x

x ln (6 x + 7) + \frac{7}{6} ln (6 x + 7) - x - \frac{7}{6} + C

Solution steps

\int ln (6 x + 7) d x

Apply u-substitution

: \int ln (u) \frac{1}{6} d u

= \int ln (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int ln (u) d u

Apply Integration By Parts

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

= \frac{1}{6} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{6} (u ln (u) - u)

Substitute back

u = 6 x + 7

= \frac{1}{6} ((6 x + 7) ln (6 x + 7) - (6 x + 7))

Simplify

\frac{1}{6} ((6 x + 7) ln (6 x + 7) - (6 x + 7)) : x ln (6 x + 7) + \frac{7}{6} ln (6 x + 7) - x - \frac{7}{6}

= x ln (6 x + 7) + \frac{7}{6} ln (6 x + 7) - x - \frac{7}{6}

Add a constant to the solution

= x ln (6 x + 7) + \frac{7}{6} ln (6 x + 7) - x - \frac{7}{6} + C

(1 + cos (t)) y^{^{'}} = (1 - e^{- y}) sin (t)

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

\frac{d}{d x} (\frac{6 x}{x ^{2} + 3})

\frac{d}{d x} (\frac{x - 7}{( x + 5 ) ^{10}})

x^{2} - 12 x + k

What is the integral of ln(6x+7) ?
The integral of ln(6x+7) is xln(6x+7)+7/6 ln(6x+7)-x-7/6+C