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Popular >

integral of y/(y-1)

Solution

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

Solution

y - 1 + ln ∣ y - 1 ∣ + C

Solution steps

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

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 = y - 1

= y - 1 + ln ∣ y - 1 ∣

Add a constant to the solution

= y - 1 + ln ∣ y - 1 ∣ + C

Popular Examples

derivative of (3x-2^6)

\frac{d}{d x} ((3 x - 2)^{6})

derivative of 1/(1+2x^2)

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

derivative of (4x^3-7)/(x^4)derivative of

\frac{4 x ^{3} - 7}{x ^{4}}

derivative of x^2+x+10

\frac{d}{d x} (x^{2} + x + 10)

derivative of f(x)=100x^5 derivative of

f (x) = 100 x^{5}

Frequently Asked Questions (FAQ)

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