What is the integral of cos(t)csc(t) ?

The integral of cos(t)csc(t) is ln|sin(t)|+C

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integral of cos(t)csc(t)

\int cos (t) csc (t) d t

ln ∣ sin (t) ∣ + C

Solution steps

\int cos (t) csc (t) d t

Rewrite using trig identities

= \int \frac{cos ( t )}{sin ( t )} d t

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = sin (t)

= ln ∣ sin (t) ∣

Add a constant to the solution

= ln ∣ sin (t) ∣ + C

What is the integral of cos(t)csc(t) ?
The integral of cos(t)csc(t) is ln|sin(t)|+C