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Popular >

integral of x/(x-1)

Solution

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

Solution

x - 1 + ln ∣ x - 1 ∣ + C

Solution steps

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

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 = x - 1

= x - 1 + ln ∣ x - 1 ∣

Add a constant to the solution

= x - 1 + ln ∣ x - 1 ∣ + C

Popular Examples

derivative of y=(e^x)/(5x^2)derivative of

y = \frac{e ^{x}}{5 x ^{2}}

derivative of (2x+3^{1.4})

\frac{d}{d x} ((2 x + 3)^{1.4})

(\partial)/(\partial x)(x+y^{1/2})

\frac{\partial}{\partial x} (x + y^{\frac{1}{2}})

limit as x approaches 0 of tan(7x)

x \to 0 lim (tan (7 x))

y^{^{'}} = y^{2} e^{2 x}

Frequently Asked Questions (FAQ)

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