What is the integral of (x+5)/(2x+3) ?

The integral of (x+5)/(2x+3) is 1/4 (2x+3+7ln|2x+3|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x+5)/(2x+3)

\int \frac{x + 5}{2 x + 3} d x

\frac{1}{4} (2 x + 3 + 7 ln ∣ 2 x + 3 ∣) + C

Solution steps

\int \frac{x + 5}{2 x + 3} d x

Apply u-substitution

: \int \frac{u + 7}{4 u} d u

= \int \frac{u + 7}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{u + 7}{u} d u

Expand

\frac{u + 7}{u} : 1 + \frac{7}{u}

= \frac{1}{4} \cdot \int 1 + \frac{7}{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}{4} (\int 1 d u + \int \frac{7}{u} d u)

\int 1 d u = u

\int \frac{7}{u} d u = 7 ln ∣ u ∣

= \frac{1}{4} (u + 7 ln ∣ u ∣)

Substitute back

u = 2 x + 3

= \frac{1}{4} (2 x + 3 + 7 ln ∣ 2 x + 3 ∣)

Add a constant to the solution

= \frac{1}{4} (2 x + 3 + 7 ln ∣ 2 x + 3 ∣) + C

\int t e^{- 8 t} d t

5 e x cos (x)

\frac{d y}{d x} = e^{x}, x (0) = 6

\frac{\partial}{\partial x} (x^{2} + y^{2} + (12 - x - y)^{2})

\frac{\partial}{\partial x} (x y e^{- y})

What is the integral of (x+5)/(2x+3) ?
The integral of (x+5)/(2x+3) is 1/4 (2x+3+7ln|2x+3|)+C