What is the integral of 15ln(x+5) ?

The integral of 15ln(x+5) is 15(xln(x+5)+5ln(x+5)-x-5)+C

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integral of 15ln(x+5)

\int 15 ln (x + 5) d x

15 (x ln (x + 5) + 5 ln (x + 5) - x - 5) + C

Solution steps

\int 15 ln (x + 5) d x

Take the constant out:

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

= 15 \cdot \int ln (x + 5) d x

Apply u-substitution

: \int ln (u) d u

= 15 \cdot \int ln (u) d u

Apply Integration By Parts

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

= 15 (u ln (u) - \int 1 d u)

\int 1 d u = u

= 15 (u ln (u) - u)

Substitute back

u = x + 5

= 15 ((x + 5) ln (x + 5) - (x + 5))

Expand

(x + 5) ln (x + 5) - (x + 5) : x ln (x + 5) + 5 ln (x + 5) - x - 5

= 15 (x ln (x + 5) + 5 ln (x + 5) - x - 5)

Add a constant to the solution

= 15 (x ln (x + 5) + 5 ln (x + 5) - x - 5) + C

\int \frac{x ^{2} + 5 x - 1}{x ^{3}} d x

\int (- 3 x + 2) e - x d x

x \to 3 lim (\frac{( x ^{2 - 7 x + 12} )}{( 3 - x )})

What is the integral of 15ln(x+5) ?
The integral of 15ln(x+5) is 15(xln(x+5)+5ln(x+5)-x-5)+C