What is the integral of 15ln(6x+10) ?

The integral of 15ln(6x+10) is 5(ln(2(3x+5))-1)(3x+5)+C

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integral of 15ln(6x+10)

\int 15 ln (6 x + 10) d x

5 (ln (2 (3 x + 5)) - 1) (3 x + 5) + C

Solution steps

\int 15 ln (6 x + 10) d x

Take the constant out:

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

= 15 \cdot \int ln (6 x + 10) d x

Apply u-substitution

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

= 15 \cdot \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

= 15 \cdot \frac{1}{6} \cdot \int ln (u) d u

Apply Integration By Parts

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

= 15 \cdot \frac{1}{6} (u ln (u) - \int 1 d u)

\int 1 d u = u

= 15 \cdot \frac{1}{6} (u ln (u) - u)

Substitute back

u = 6 x + 10

= 15 \cdot \frac{1}{6} ((6 x + 10) ln (6 x + 10) - (6 x + 10))

Simplify

15 \cdot \frac{1}{6} ((6 x + 10) ln (6 x + 10) - (6 x + 10)) : 5 (ln (2 (3 x + 5)) - 1) (3 x + 5)

= 5 (ln (2 (3 x + 5)) - 1) (3 x + 5)

Add a constant to the solution

= 5 (ln (2 (3 x + 5)) - 1) (3 x + 5) + C

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What is the integral of 15ln(6x+10) ?
The integral of 15ln(6x+10) is 5(ln(2(3x+5))-1)(3x+5)+C