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

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

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

integral

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

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

Solution steps

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

Apply u-substitution

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

= \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 v + 22

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

Add a constant to the solution

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

\frac{x ^{2}}{( a ^{2} + x ^{2} ) ^{2}}

\frac{x ^{3}}{x - 2}

\frac{e ^{3}}{x}

\frac{x ^{0.5}}{( 1 + x ) ^{2}}

\frac{x}{( ( x - 1 ) ( x - 2 ) ( x - 3 ) )}

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