What is the integral of csc^2(a-bx) ?

The integral of csc^2(a-bx) is 1/b cot(a-bx)+C

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integral of csc^2(a-bx)

\int csc^{2} (a - b x) d x

\frac{1}{b} cot (a - b x) + C

Solution steps

\int csc^{2} (a - b x) d x

Apply u-substitution

: \int - \frac{csc ^{2} ( u )}{b} d u

= \int - \frac{csc ^{2} ( u )}{b} d u

Take the constant out:

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

= - \frac{1}{b} \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= - \frac{1}{b} (- cot (u))

Substitute back

u = a - b x

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

Simplify

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

Add a constant to the solution

= \frac{1}{b} cot (a - b x) + C

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What is the integral of csc^2(a-bx) ?
The integral of csc^2(a-bx) is 1/b cot(a-bx)+C