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

The integral of 1/(sqrt(1-cos(2x))) is 1/(sqrt(2))ln|tan(x/2)|+C

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

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

\frac{1}{2} ln tan (\frac{x}{2}) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{sin ^{2} ( x )} d x

sin^{2} (x) = (sin (x)),

assuming

sin (x) \geq 0

= \frac{1}{2} \cdot \int \frac{1}{sin ( x )} d x

Apply u-substitution

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = tan (\frac{x}{2})

= \frac{1}{2} ln tan (\frac{x}{2})

Add a constant to the solution

= \frac{1}{2} ln tan (\frac{x}{2}) + C

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What is the integral of 1/(sqrt(1-cos(2x))) ?
The integral of 1/(sqrt(1-cos(2x))) is 1/(sqrt(2))ln|tan(x/2)|+C