What is the integral of csc(9x) ?

The integral of csc(9x) is 1/9 ln|tan((9x)/2)|+C

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integral of csc(9x)

\int csc (9 x) d x

\frac{1}{9} ln tan (\frac{9 x}{2}) + C

Solution steps

\int csc (9 x) d x

Apply u-substitution

: \int csc (u) \frac{1}{9} d u

= \int csc (u) \frac{1}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int csc (u) d u

Apply u-substitution

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

= \frac{1}{9} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{9} ln ∣ v ∣

Substitute back

= \frac{1}{9} ln tan (\frac{9 x}{2})

Add a constant to the solution

= \frac{1}{9} ln tan (\frac{9 x}{2}) + C

\int \frac{3 x}{x + 4} d x

\int y^{2} + 4 d y

x sin (x)

\frac{\partial}{\partial x} (x + x^{2} + y^{2})

\frac{d}{d x} (4 x - 2)

What is the integral of csc(9x) ?
The integral of csc(9x) is 1/9 ln|tan((9x)/2)|+C