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Popular >

integral of y/(y-2)

Solution

\int \frac{y}{y - 2} d y

Solution

y - 2 + 2 ln ∣ y - 2 ∣ + C

Solution steps

\int \frac{y}{y - 2} d y

Apply u-substitution

: \int \frac{u + 2}{u} d u

= \int \frac{u + 2}{u} d u

Expand

\frac{u + 2}{u} : 1 + \frac{2}{u}

= \int 1 + \frac{2}{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{2}{u} d u

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= u + 2 ln ∣ u ∣

Substitute back

u = y - 2

= y - 2 + 2 ln ∣ y - 2 ∣

Add a constant to the solution

= y - 2 + 2 ln ∣ y - 2 ∣ + C

Popular Examples

integral of 3z^2ln(z)

\int 3 z^{2} ln (z) d z

integral of r^3cos^2(θ)

\int r^{3} cos^{2} (θ) d r

integral of (x^3+3x)/(x^2-3x-10)

\int \frac{x ^{3} + 3 x}{x ^{2} - 3 x - 10} d x

integral of 1/(x^{2-4)}

\int \frac{1}{x ^{2 - 4}} d x

integral of x^4*e^{(1-x^5)}

\int x^{4} \cdot e^{(1 - x^{5})} d x

Frequently Asked Questions (FAQ)

What is the integral of y/(y-2) ?
The integral of y/(y-2) is y-2+2ln|y-2|+C