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

The integral of x/((2+5x)) is 1/25 (2+5x-2ln|2+5x|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

integral

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

\frac{1}{25} (2 + 5 x - 2 ln ∣ 2 + 5 x ∣) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{25 u} d u

= \int \frac{u - 2}{25 u} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{u - 2}{u} d u

Expand

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

= \frac{1}{25} \cdot \int 1 - \frac{2}{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}{25} (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{25} (u - 2 ln ∣ u ∣)

Substitute back

u = 2 + 5 x

= \frac{1}{25} (2 + 5 x - 2 ln ∣ 2 + 5 x ∣)

Add a constant to the solution

= \frac{1}{25} (2 + 5 x - 2 ln ∣ 2 + 5 x ∣) + C

\frac{sin ^{2} ( x ) tan ^{2} ( x )}{2}

\frac{x}{x - 1}

\frac{1}{( 2 x )}

\frac{ln ( x )}{( 1 + 2 x ) ^{2}}

\frac{1}{( 100 - n )}

What is the integral of x/((2+5x)) ?
The integral of x/((2+5x)) is 1/25 (2+5x-2ln|2+5x|)+C