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Popular >

integral of x/(x-2)

Solution

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

Solution

x - 2 + 2 ln ∣ x - 2 ∣ + C

Solution steps

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

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 = x - 2

= x - 2 + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= x - 2 + 2 ln ∣ x - 2 ∣ + C

Popular Examples

y^'=(x-2)/(y+1)

y^{^{'}} = \frac{x - 2}{y + 1}

integral of-|x|^2

\int - ∣ x ∣^{2} d x

integral from 0 to 1 of 4ln(x)

\int_{0}^{1} 4 ln (x) d x

derivative of (sin^2(x)/2)

\frac{d}{d x} (\frac{sin ^{2} ( x )}{2})

slope derivative of 5/(4x)-7/4 slope

\frac{d}{d x} (\frac{5}{4 x}) - \frac{7}{4}

Frequently Asked Questions (FAQ)

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