What is the integral of csc(1-5x) ?

The integral of csc(1-5x) is -1/5 ln|tan((1-5x)/2)|+C

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integral of csc(1-5x)

\int csc (1 - 5 x) d x

- \frac{1}{5} ln tan (\frac{1 - 5 x}{2}) + C

Solution steps

\int csc (1 - 5 x) d x

Apply u-substitution

: \int - \frac{1}{5} csc (u) d u

= \int - \frac{1}{5} csc (u) d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int csc (u) d u

Apply u-substitution

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

= - \frac{1}{5} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{5} ln ∣ v ∣

Substitute back

= - \frac{1}{5} ln tan (\frac{1 - 5 x}{2})

Add a constant to the solution

= - \frac{1}{5} ln tan (\frac{1 - 5 x}{2}) + C

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What is the integral of csc(1-5x) ?
The integral of csc(1-5x) is -1/5 ln|tan((1-5x)/2)|+C