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Popular >

integral of v/(v+1)

Solution

\int \frac{v}{v + 1} d v

Solution

v + 1 - ln ∣ v + 1 ∣ + C

Solution steps

\int \frac{v}{v + 1} d v

Apply u-substitution

: \int \frac{u - 1}{u} d u

= \int \frac{u - 1}{u} d u

Expand

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

= \int 1 - \frac{1}{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

= \int 1 d u - \int \frac{1}{u} d u

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = v + 1

= v + 1 - ln ∣ v + 1 ∣

Add a constant to the solution

= v + 1 - ln ∣ v + 1 ∣ + C

Popular Examples

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\int 12 sin (2 x) d x

integral of 1/(x(1-2ln(x)))

\int \frac{1}{x ( 1 - 2 ln ( x ) )} d x

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\int (p^{- 1} + \frac{4}{x}) d x

integral of x*sqrt(x+2)+2*sqrt(x+2)

\int x \cdot x + 2 + 2 \cdot x + 2 d x

Frequently Asked Questions (FAQ)

What is the integral of v/(v+1) ?
The integral of v/(v+1) is v+1-ln|v+1|+C