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Popular >

integral of v/(1-v)

Solution

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

Solution

1 - v - ln ∣ 1 - v ∣ + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= - \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 = 1 - v

= - (- (1 - v) + ln ∣ 1 - v ∣)

Simplify

= 1 - v - ln ∣ 1 - v ∣

Add a constant to the solution

= 1 - v - ln ∣ 1 - v ∣ + C

Popular Examples

integral of-x+3(cos((n*pi*x)/8))

\int - x + 3 (cos (\frac{n \cdot π \cdot x}{8})) d x

derivative of 1/(sqrt(xy))

\frac{d}{d x} (\frac{1}{x y})

(x^2-xy)(dy)/(dx)+y^2=0

(x^{2} - x y) \frac{d y}{d x} + y^{2} = 0

y = t

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

\int \frac{2 x - 1}{x ^{2}} d x

Frequently Asked Questions (FAQ)

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