What is the integral of cot(θ/9) ?

The integral of cot(θ/9) is 9ln|sin(θ/9)|+C

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integral of cot(θ/9)

\int cot (\frac{θ}{9}) d θ

9 ln sin (\frac{θ}{9}) + C

Solution steps

\int cot (\frac{θ}{9}) d θ

Rewrite using trig identities

= \int \frac{cos ( \frac{θ}{9} )}{sin ( \frac{θ}{9} )} d θ

Apply u-substitution

: \int \frac{9}{u} d u

= \int \frac{9}{u} d u

Take the constant out:

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

= 9 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 ln ∣ u ∣

Substitute back

u = sin (\frac{θ}{9})

= 9 ln sin (\frac{θ}{9})

Add a constant to the solution

= 9 ln sin (\frac{θ}{9}) + C

x \to 0 lim (\frac{∣ x - 5 ∣}{x - 5})

\frac{- x ^{2} - 5}{( x ^{2} - 5 ) ^{2}}

y^{^{'}} + 4 y = 20, y (1) = 30

\frac{d y}{d x} = 6 x - y

f (f (x))