What is the integral of csc^2((x-1)/3) ?

The integral of csc^2((x-1)/3) is -3cot(1/3 (x-1))+C

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integral of csc^2((x-1)/3)

\int csc^{2} (\frac{x - 1}{3}) d x

- 3 cot (\frac{1}{3} (x - 1)) + C

Solution steps

\int csc^{2} (\frac{x - 1}{3}) d x

Apply u-substitution

: \int csc^{2} (\frac{u}{3}) d u

= \int csc^{2} (\frac{u}{3}) d u

Rewrite using trig identities

= \int 1 + cot^{2} (\frac{u}{3}) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int cot^{2} (\frac{u}{3}) d u

\int 1 d u = u

\int cot^{2} (\frac{u}{3}) d u = - u - 3 cot (\frac{u}{3})

= u - u - 3 cot (\frac{u}{3})

Substitute back

u = x - 1

= x - 1 - (x - 1) - 3 cot (\frac{x - 1}{3})

Simplify

x - 1 - (x - 1) - 3 cot (\frac{x - 1}{3}) : - 3 cot (\frac{1}{3} (x - 1))

= - 3 cot (\frac{1}{3} (x - 1))

Add a constant to the solution

= - 3 cot (\frac{1}{3} (x - 1)) + C

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What is the integral of csc^2((x-1)/3) ?
The integral of csc^2((x-1)/3) is -3cot(1/3 (x-1))+C