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Popular >

integral of t/(1-t)

Solution

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

Solution

1 - t - ln ∣ 1 - t ∣ + C

Solution steps

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

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 - t

= - (- (1 - t) + ln ∣ 1 - t ∣)

Simplify

= 1 - t - ln ∣ 1 - t ∣

Add a constant to the solution

= 1 - t - ln ∣ 1 - t ∣ + C

Popular Examples

limit as x approaches-3 of-2x-3

x \to - 3 lim (- 2 x - 3)

integral of \sqrt[17]{cot(x)}csc^2(x)

limit as x approaches 0 of (ln(1+x^2))/x

x \to 0 lim (\frac{ln ( 1 + x ^{2} )}{x})

(d^2)/(dx^2)((x^2+4)/(x^2-4))

\frac{d ^{2}}{d x ^{2}} (\frac{x ^{2} + 4}{x ^{2} - 4})

derivative of 8x+5

\frac{d}{d x} (8 x + 5)

Frequently Asked Questions (FAQ)

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