What is the integral of 3/(sqrt(x)(1-cos(\sqrt{x)))} ?

The integral of 3/(sqrt(x)(1-cos(\sqrt{x)))} is -6cot(1/2 sqrt(x))+C

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integral of 3/(sqrt(x)(1-cos(\sqrt{x)))}

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

- 6 cot (\frac{1}{2} x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 3 \cdot \int \frac{2}{1 - cos ( u )} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{1}{1 - cos ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= 3 \cdot 2 \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= 3 \cdot 2 (- \frac{1}{v})

Substitute back

= 3 \cdot 2 - \frac{1}{tan ( \frac{x}{2} )}

Simplify

3 \cdot 2 - \frac{1}{tan ( \frac{x}{2} )} : - 6 cot (\frac{1}{2} x)

= - 6 cot (\frac{1}{2} x)

Add a constant to the solution

= - 6 cot (\frac{1}{2} x) + C

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\int \frac{x ^{3} - 1}{x ^{2} + 1} d x

\frac{d}{d x} (cot^{4} (x))

What is the integral of 3/(sqrt(x)(1-cos(\sqrt{x)))} ?
The integral of 3/(sqrt(x)(1-cos(\sqrt{x)))} is -6cot(1/2 sqrt(x))+C