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Popular >

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

Popular Examples

integral from 0 to 2.1 of pi(21/10-x)^2

\int_{0}^{2.1} π (\frac{21}{10} - x)^{2} d x

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

\frac{\partial ^{2}}{\partial y ^{2}} (ln (2 + x^{2} y^{2}))

integral of tan^3(x)sec^6(x)

\int tan^{3} (x) sec^{6} (x) d x

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

\int \frac{1}{9} x^{- \frac{2}{9}} d x

(\partial)/(\partial x)(y^2*x)

\frac{\partial}{\partial x} (y^{2} \cdot x)

Frequently Asked Questions (FAQ)

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