What is the integral of 1/(sqrt((x-1)(3-x))) ?

The integral of 1/(sqrt((x-1)(3-x))) is arcsin(x-2)+C

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integral of 1/(sqrt((x-1)(3-x)))

\int \frac{1}{( x - 1 ) ( 3 - x )} d x

arcsin (x - 2) + C

Solution steps

\int \frac{1}{( x - 1 ) ( 3 - x )} d x

Complete the square

(x - 1) (3 - x) : - (x - 2)^{2} + 1

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

Apply u-substitution

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

= \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 - 2

= arcsin (x - 2)

Add a constant to the solution

= arcsin (x - 2) + C

\int (\frac{1}{1 + y ^{2}}) d y

\int \frac{4 x ^{3}}{x ^{2} - 16} d x

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\int 1 - 3 x d x

What is the integral of 1/(sqrt((x-1)(3-x))) ?
The integral of 1/(sqrt((x-1)(3-x))) is arcsin(x-2)+C