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Popular >

integral of z/(1-z)

Solution

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

Solution

1 - z - ln ∣ 1 - z ∣ + C

Solution steps

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

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 - z

= - (- (1 - z) + ln ∣ 1 - z ∣)

Simplify

= 1 - z - ln ∣ 1 - z ∣

Add a constant to the solution

= 1 - z - ln ∣ 1 - z ∣ + C

Popular Examples

integral of 0.16e^{-0.16x}

\int 0.16 e^{- 0.16 x} d x

integral of 5xsqrt(1^2+5^2)

\int 5 x 1^{2} + 5^{2} d x

integral of x^2*sqrt(2x^3+1)

\int x^{2} \cdot 2 x^{3} + 1 d x

integral of (e^x+ln(y)+y/x)

\int (e^{x} + ln (y) + \frac{y}{x}) d x

integral of (x^3-x)^2

\int (x^{3} - x)^{2} d x

Frequently Asked Questions (FAQ)

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