What is the integral of csc^2((x+8)/4) ?

The integral of csc^2((x+8)/4) is -4cot(1/4 (x+8))+C

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integral of csc^2((x+8)/4)

\int csc^{2} (\frac{x + 8}{4}) d x

- 4 cot (\frac{1}{4} (x + 8)) + C

Solution steps

\int csc^{2} (\frac{x + 8}{4}) d x

Apply u-substitution

: \int csc^{2} (\frac{u}{4}) d u

= \int csc^{2} (\frac{u}{4}) d u

Rewrite using trig identities

= \int 1 + cot^{2} (\frac{u}{4}) 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}{4}) d u

\int 1 d u = u

\int cot^{2} (\frac{u}{4}) d u = - u - 4 cot (\frac{u}{4})

= u - u - 4 cot (\frac{u}{4})

Substitute back

u = x + 8

= x + 8 - (x + 8) - 4 cot (\frac{x + 8}{4})

Simplify

x + 8 - (x + 8) - 4 cot (\frac{x + 8}{4}) : - 4 cot (\frac{1}{4} (x + 8))

= - 4 cot (\frac{1}{4} (x + 8))

Add a constant to the solution

= - 4 cot (\frac{1}{4} (x + 8)) + C

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What is the integral of csc^2((x+8)/4) ?
The integral of csc^2((x+8)/4) is -4cot(1/4 (x+8))+C