What is the integral of csc((ax)/b)*cot((ax)/b) ?

The integral of csc((ax)/b)*cot((ax)/b) is -b/a csc((ax)/b)+C

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integral of csc((ax)/b)*cot((ax)/b)

\int csc (\frac{a x}{b}) \cdot cot (\frac{a x}{b}) d x

- \frac{b}{a} csc (\frac{a x}{b}) + C

Solution steps

\int csc (\frac{a x}{b}) cot (\frac{a x}{b}) d x

Apply u-substitution

: \int \frac{b csc ( u ) cot ( u )}{a} d u

= \int \frac{b csc ( u ) cot ( u )}{a} d u

Take the constant out:

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

= \frac{b}{a} \cdot \int csc (u) cot (u) d u

Apply u-substitution

: \int - 1 d v

= \frac{b}{a} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= \frac{b}{a} (- 1) v

Substitute back

= \frac{b}{a} (- 1) csc (\frac{a x}{b})

Simplify

= - \frac{b}{a} csc (\frac{a x}{b})

Add a constant to the solution

= - \frac{b}{a} csc (\frac{a x}{b}) + C

f (x) = 4 x - 3

\int_{0}^{2} e^{- 2 x} + 4 d x

\frac{d}{d x} (\frac{6}{2 x + 1})

tan^{3} (x)

\frac{d}{d x} (\frac{150}{x})

What is the integral of csc((ax)/b)*cot((ax)/b) ?
The integral of csc((ax)/b)*cot((ax)/b) is -b/a csc((ax)/b)+C