What is the integral of (2v)/(7v-12) ?

The integral of (2v)/(7v-12) is 2/49 (7v-12+12ln|7v-12|)+C

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integral of (2v)/(7v-12)

\int \frac{2 v}{7 v - 12} d v

\frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + C

Solution steps

\int \frac{2 v}{7 v - 12} d v

Take the constant out:

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

= 2 \cdot \int \frac{v}{7 v - 12} d v

Apply u-substitution

: \int \frac{u + 12}{49 u} d u

= 2 \cdot \int \frac{u + 12}{49 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{49} \cdot \int \frac{u + 12}{u} d u

Expand

\frac{u + 12}{u} : 1 + \frac{12}{u}

= 2 \cdot \frac{1}{49} \cdot \int 1 + \frac{12}{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

= 2 \cdot \frac{1}{49} (\int 1 d u + \int \frac{12}{u} d u)

\int 1 d u = u

\int \frac{12}{u} d u = 12 ln ∣ u ∣

= 2 \cdot \frac{1}{49} (u + 12 ln ∣ u ∣)

Substitute back

u = 7 v - 12

= 2 \cdot \frac{1}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

Simplify

2 \cdot \frac{1}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) : \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

= \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

Add a constant to the solution

= \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + C

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What is the integral of (2v)/(7v-12) ?
The integral of (2v)/(7v-12) is 2/49 (7v-12+12ln|7v-12|)+C