What is the integral of ln(5-x) ?

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

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integral of ln(5-x)

\int ln (5 - x) d x

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

Solution steps

\int ln (5 - x) d x

Apply u-substitution

: \int - ln (u) d u

= \int - ln (u) d u

Take the constant out:

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

= - \int ln (u) d u

Apply Integration By Parts

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

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

\int 1 d u = u

= - (u ln (u) - u)

Substitute back

u = 5 - x

= - ((5 - x) ln (5 - x) - (5 - x))

Simplify

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

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

Add a constant to the solution

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

f (x) = 2 x^{2} - 7 x + 5

\int \frac{x ^{3} + 2 x ^{2} - 18 x + 2}{x ^{2} + x - 12} d x

(x^{2} + 1) \frac{d y}{d x} + 3 x (y - 1) = 0, y (0) = 7

x \to 1 lim (- 2 x + 3)

(t + 2)^{2}

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