What is the integral of 1/(sqrt(x(2-x))) ?

The integral of 1/(sqrt(x(2-x))) is arcsin(x-1)+C

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integral of 1/(sqrt(x(2-x)))

\int \frac{1}{x ( 2 - x )} d x

arcsin (x - 1) + C

Solution steps

\int \frac{1}{x ( 2 - x )} d x

Complete the square

x (2 - x) : - (x - 1)^{2} + 1

= \int \frac{1}{- ( x - 1 ) ^{2} + 1} d x

Apply u-substitution

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

Use the common integral:

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

= arcsin (u)

Substitute back

u = x - 1

= arcsin (x - 1)

Add a constant to the solution

= arcsin (x - 1) + C

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What is the integral of 1/(sqrt(x(2-x))) ?
The integral of 1/(sqrt(x(2-x))) is arcsin(x-1)+C