What is the integral of 5/(x-4) ?

The integral of 5/(x-4) is 5ln|x-4|+C

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integral of 5/(x-4)

\int \frac{5}{x - 4} d x

5 ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{5}{x - 4} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x - 4} d x

Apply u-substitution

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

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = x - 4

= 5 ln ∣ x - 4 ∣

Add a constant to the solution

= 5 ln ∣ x - 4 ∣ + C

3 x \frac{d y}{d x} + y = 0, y (1) = 1

(x, y) \to (in f ini t y, in f ini t y) lim (x y)

y = 8 x + 9 cos (x), (0, 9)

\int \frac{e ^{x}}{x ^{2}} d x

\frac{d}{d x} (\frac{e ^{2 x - 1}}{( x ^{2} + 5 x )})