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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

integral of (tan^3(ln(x)))/(5x)

\int \frac{tan ^{3} ( ln ( x ) )}{5 x} d x

y'=(xy+y^2)/(x^2)

y^{'} = \frac{x y + y ^{2}}{x ^{2}}

integral of (e^{3x})/(4-9e^{6x)}

\int \frac{e ^{3 x}}{4 - 9 e ^{6 x}} d x

y^{^{'}} - \frac{y}{x} = y^{2}

sqrt(xy)((dy)/(dx))=3

x y (\frac{d y}{d x}) = 3

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