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

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

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integral of csc^2(x/4)

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

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

Solution steps

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

Rewrite using trig identities

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

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

\int 1 d x = x

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

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

Simplify

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

Add a constant to the solution

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

y = \frac{x}{3} (2 x + 1)^{3}

\int \frac{e ^{3 s i n (x)}}{sec ( x )} d x

f (x) = x^{2 x}

\int \frac{- ln ( x )}{x} d x

y = c e^{x}