What is the integral of-2cot(y) ?

The integral of-2cot(y) is -2ln|sin(y)|+C

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integral of-2cot(y)

\int - 2 cot (y) d y

- 2 ln ∣ sin (y) ∣ + C

Solution steps

\int - 2 cot (y) d y

Take the constant out:

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

= - 2 \cdot \int cot (y) d y

Rewrite using trig identities

= - 2 \cdot \int \frac{cos ( y )}{sin ( y )} d y

Apply u-substitution

: \int \frac{1}{u} d u

= - 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2 ln ∣ u ∣

Substitute back

u = sin (y)

= - 2 ln ∣ sin (y) ∣

Add a constant to the solution

= - 2 ln ∣ sin (y) ∣ + C

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