What is the integral of csc(-3h) ?

The integral of csc(-3h) is -1/3 ln|tan((3h)/2)|+C

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integral of csc(-3h)

\int csc (- 3 h) d h

- \frac{1}{3} ln tan (\frac{3 h}{2}) + C

Solution steps

\int csc (- 3 h) d h

Apply u-substitution

= \int - \frac{1}{3} csc (u) d u

Take the constant out:

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

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

Apply u-substitution

= - \frac{1}{3} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{3} ln ∣ v ∣

Substitute back

= - \frac{1}{3} ln tan (\frac{- 3 h}{2})

Simplify

- \frac{1}{3} ln tan (\frac{- 3 h}{2}) : - \frac{1}{3} ln tan (\frac{3 h}{2})

= - \frac{1}{3} ln tan (\frac{3 h}{2})

Add a constant to the solution

= - \frac{1}{3} ln tan (\frac{3 h}{2}) + C

What is the integral of csc(-3h) ?
The integral of csc(-3h) is -1/3 ln|tan((3h)/2)|+C