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Popular Calculus >

integral of (1-y)/y

Solution

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

Solution

ln ∣ y ∣ - y + C

Solution steps

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

Expand

\frac{1 - y}{y} : \frac{1}{y} - 1

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{y} d y = ln ∣ y ∣

\int 1 d y = y

= ln ∣ y ∣ - y

Add a constant to the solution

= ln ∣ y ∣ - y + C

Graph

Popular Examples

(\partial ^2)/(\partial y^2)(ln(2+x^2y^2))

integral of 1/9 x^{-2/9}

integral from 1 to 9 of 1/(2x^2)

x^2y^{''}+2xy^'-12y=0,y(1)=-1,y^'(1)=10

(dx)/(dy)=-(4y^2+6xy)/(3y^2+2x)

Frequently Asked Questions (FAQ)

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