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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

ln ∣ x + 5 ∣ + \frac{5}{x + 5} + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 5}{u ^{2}} d u

= \int \frac{u - 5}{u ^{2}} d u

Expand

\frac{u - 5}{u ^{2}} : \frac{1}{u} - \frac{5}{u ^{2}}

= \int \frac{1}{u} - \frac{5}{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{1}{u} d u - \int \frac{5}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{5}{u ^{2}} d u = - \frac{5}{u}

= ln ∣ u ∣ - (- \frac{5}{u})

Substitute back

u = x + 5

= ln ∣ x + 5 ∣ - (- \frac{5}{x + 5})

Simplify

= ln ∣ x + 5 ∣ + \frac{5}{x + 5}

Add a constant to the solution

= ln ∣ x + 5 ∣ + \frac{5}{x + 5} + C

\frac{d}{d x} ((12 x^{2} - 24 x + 24) e^{x})

\frac{d}{d x} ((\frac{2}{3})^{x})

f (x) = (4 x - 3) (x + 2)

f (x) = x^{2} + 10

\int sec^{2} (x) tan^{10} (x) d x

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