What is the integral of (4x+8)/((x+5)^2) ?

The integral of (4x+8)/((x+5)^2) is 4ln|x+5|+(12)/(x+5)+C

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integral of (4x+8)/((x+5)^2)

\int \frac{4 x + 8}{( x + 5 ) ^{2}} d x

4 ln ∣ x + 5 ∣ + \frac{12}{x + 5} + C

Solution steps

\int \frac{4 x + 8}{( x + 5 ) ^{2}} d x

Apply u-substitution

: \int \frac{4 u - 12}{u ^{2}} d u

= \int \frac{4 u - 12}{u ^{2}} d u

Expand

\frac{4 u - 12}{u ^{2}} : \frac{4}{u} - \frac{12}{u ^{2}}

= \int \frac{4}{u} - \frac{12}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4}{u} d u - \int \frac{12}{u ^{2}} d u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

= 4 ln ∣ u ∣ - (- \frac{12}{u})

Substitute back

u = x + 5

= 4 ln ∣ x + 5 ∣ - (- \frac{12}{x + 5})

Simplify

= 4 ln ∣ x + 5 ∣ + \frac{12}{x + 5}

Add a constant to the solution

= 4 ln ∣ x + 5 ∣ + \frac{12}{x + 5} + C

\frac{1}{4 x ^{\frac{3}{4}}}

(- 7.4) (- 4.1)

- sin (x)

(\frac{2}{2 s + 1}) - (\frac{1}{s})

(y^{2} + x y) d x - x^{2} d y = 0

What is the integral of (4x+8)/((x+5)^2) ?
The integral of (4x+8)/((x+5)^2) is 4ln|x+5|+(12)/(x+5)+C