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Popular >

integral of x/(1-x)

Solution

integral

\frac{x}{1 - x}

Solution

1 - x - ln ∣ 1 - x ∣ + C

Solution steps

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

Apply u-substitution

: \int - \frac{- u + 1}{u} d u

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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 = 1 - x

= - (- (1 - x) + ln ∣ 1 - x ∣)

Simplify

= 1 - x - ln ∣ 1 - x ∣

Add a constant to the solution

= 1 - x - ln ∣ 1 - x ∣ + C

Popular Examples

integral of (-x)/((1+x))integral

\frac{- x}{( 1 + x )}

integral of (1-t^2)^3 integral

(1 - t^{2})^{3}

integral of (5x^4)/(sqrt(x^5+9))integral

\frac{5 x ^{4}}{x ^{5} + 9}

integral of x*log_{10}(1/x)integral

x \cdot lo g_{10} (\frac{1}{x})

integral of 1/((x(sqrt(ln(x)))))integral

\frac{1}{( x ( ln ( x ) ) )}

Frequently Asked Questions (FAQ)

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