What is the integral of ((2x+5))/((7x+22)) ?

The integral of ((2x+5))/((7x+22)) is 1/49 (2(7x+22)-9ln|7x+22|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ((2x+5))/((7x+22))

integral

\frac{( 2 x + 5 )}{( 7 x + 22 )}

\frac{1}{49} (2 (7 x + 22) - 9 ln ∣ 7 x + 22 ∣) + C

Solution steps

\int \frac{2 x + 5}{7 x + 22} d x

Apply u-substitution

= \int \frac{2 u - 9}{49 u} d u

Take the constant out:

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

= \frac{1}{49} \cdot \int \frac{2 u - 9}{u} d u

Expand

\frac{2 u - 9}{u} : 2 - \frac{9}{u}

= \frac{1}{49} \cdot \int 2 - \frac{9}{u} 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

= \frac{1}{49} (\int 2 d u - \int \frac{9}{u} d u)

\int 2 d u = 2 u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

= \frac{1}{49} (2 u - 9 ln ∣ u ∣)

Substitute back

u = 7 x + 22

= \frac{1}{49} (2 (7 x + 22) - 9 ln ∣ 7 x + 22 ∣)

Add a constant to the solution

= \frac{1}{49} (2 (7 x + 22) - 9 ln ∣ 7 x + 22 ∣) + C

What is the integral of ((2x+5))/((7x+22)) ?
The integral of ((2x+5))/((7x+22)) is 1/49 (2(7x+22)-9ln|7x+22|)+C