What is the integral of 1/(tsqrt(1-t)) ?

The integral of 1/(tsqrt(1-t)) is -ln|sqrt(1-t)+1|+ln|sqrt(1-t)-1|+C

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integral of 1/(tsqrt(1-t))

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

- ln 1 - t + 1 + ln 1 - t - 1 + C

Solution steps

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

Apply u-substitution

: \int - \frac{2}{- u ^{2} + 1} d u

= \int - \frac{2}{- u ^{2} + 1} d u

Take the constant out:

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

= - 2 \cdot \int \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= - 2 (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = 1 - t

= - 2 (\frac{ln 1 - t + 1}{2} - \frac{ln 1 - t - 1}{2})

Simplify

- 2 (\frac{ln 1 - t + 1}{2} - \frac{ln 1 - t - 1}{2}) : - ln 1 - t + 1 + ln 1 - t - 1

= - ln 1 - t + 1 + ln 1 - t - 1

Add a constant to the solution

= - ln 1 - t + 1 + ln 1 - t - 1 + C

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