What is the integral of (8x)/(2x+7) ?

The integral of (8x)/(2x+7) is 2(2x+7-7ln|2x+7|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (8x)/(2x+7)

\int \frac{8 x}{2 x + 7} d x

2 (2 x + 7 - 7 ln ∣ 2 x + 7 ∣) + C

Solution steps

\int \frac{8 x}{2 x + 7} d x

Take the constant out:

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

= 8 \cdot \int \frac{x}{2 x + 7} d x

Apply u-substitution

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

= 8 \cdot \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

= 8 \cdot \frac{1}{4} \cdot \int \frac{u - 7}{u} d u

Expand

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

= 8 \cdot \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

= 8 \cdot \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 ∣

= 8 \cdot \frac{1}{4} (u - 7 ln ∣ u ∣)

Substitute back

u = 2 x + 7

= 8 \cdot \frac{1}{4} (2 x + 7 - 7 ln ∣ 2 x + 7 ∣)

Simplify

8 \cdot \frac{1}{4} (2 x + 7 - 7 ln ∣ 2 x + 7 ∣) : 2 (2 x + 7 - 7 ln ∣ 2 x + 7 ∣)

= 2 (2 x + 7 - 7 ln ∣ 2 x + 7 ∣)

Add a constant to the solution

= 2 (2 x + 7 - 7 ln ∣ 2 x + 7 ∣) + C

\int \frac{2 e ^{x}}{e ^{3 x} + 2 e ^{x}} d x

\int (x^{- 1} + 3)^{3} d x

\int sec^{2} (a x) d x

\int + x^{3} ln (5 x) d x

\int \frac{1}{1000} e^{- \frac{x}{1000}} d x

What is the integral of (8x)/(2x+7) ?
The integral of (8x)/(2x+7) is 2(2x+7-7ln|2x+7|)+C