What is the integral of a/(a-x) ?

The integral of a/(a-x) is -aln|a-x|+C

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integral of a/(a-x)

\int \frac{a}{a - x} d x

- a ln ∣ a - x ∣ + C

Solution steps

\int \frac{a}{a - x} d x

Take the constant out:

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

= a \cdot \int \frac{1}{a - x} d x

Apply u-substitution

: \int - \frac{1}{u} d u

= a \cdot \int - \frac{1}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a (- ln ∣ u ∣)

Substitute back

u = a - x

= a (- ln ∣ a - x ∣)

Simplify

= - a ln ∣ a - x ∣

Add a constant to the solution

= - a ln ∣ a - x ∣ + C

y^{^{'}} = \frac{( y ^{2} - 1 )}{x y}

\frac{d}{d x} (x e^{x^{4}})

\frac{d}{d z} (x^{2} y z)

\int x^{5} \cdot ln (x) d x

y^{^{''}} + 9 y = 19