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

The integral of (csc(sqrt(2x)))/(sqrt(x)) is sqrt(2)ln|tan(sqrt(x/2))|+C

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integral of (csc(sqrt(2x)))/(sqrt(x))

\int \frac{csc ( 2 x )}{x} d x

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

Solution steps

\int \frac{csc ( 2 x )}{x} d x

Apply u-substitution

: \int 2 csc (u) d u

= \int 2 csc (u) d u

Take the constant out:

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

= 2 \cdot \int csc (u) d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 ln ∣ v ∣

Substitute back

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

\frac{2 x}{2} = \frac{x}{2}

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

Add a constant to the solution

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

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