What is the integral of cot(x/5) ?

The integral of cot(x/5) is 5ln|sin(x/5)|+C

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integral of cot(x/5)

\int cot (\frac{x}{5}) d x

5 ln sin (\frac{x}{5}) + C

Solution steps

\int cot (\frac{x}{5}) d x

Rewrite using trig identities

= \int \frac{cos ( \frac{x}{5} )}{sin ( \frac{x}{5} )} d x

Apply u-substitution

: \int \frac{5}{u} d u

= \int \frac{5}{u} d u

Take the constant out:

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

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = sin (\frac{x}{5})

= 5 ln sin (\frac{x}{5})

Add a constant to the solution

= 5 ln sin (\frac{x}{5}) + C

y = 3 - x^{2}, y = 7 - 5 x

x \to 1 lim (\frac{x ^{2} - 9}{x + 3})

\frac{d}{d x} (∣ f (x) ∣)

x \to \frac{π}{4} lim (csc (x))

\int x 5 + x d x