What is the integral of (csc^2(t))/(cot(t)) ?

The integral of (csc^2(t))/(cot(t)) is -ln|cot(t)|+C

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integral of (csc^2(t))/(cot(t))

\int \frac{csc ^{2} ( t )}{cot ( t )} d t

- ln ∣ cot (t) ∣ + C

Solution steps

\int \frac{csc ^{2} ( t )}{cot ( t )} d t

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cot (t)

= - ln ∣ cot (t) ∣

Add a constant to the solution

= - ln ∣ cot (t) ∣ + C

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What is the integral of (csc^2(t))/(cot(t)) ?
The integral of (csc^2(t))/(cot(t)) is -ln|cot(t)|+C