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

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

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

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

ln tan (\frac{t}{2}) + cos (t) + C

Solution steps

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

Simplify

\frac{cot ^{2} ( t )}{csc ( t )} : cos (t) cot (t)

= \int cos (t) cot (t) d t

Rewrite using trig identities

= \int \frac{1 - sin ^{2} ( t )}{sin ( t )} d t

Expand

\frac{1 - sin ^{2} ( t )}{sin ( t )} : \frac{1}{sin ( t )} - sin (t)

= \int \frac{1}{sin ( t )} - sin (t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{sin ( t )} d t - \int sin (t) d t

\int \frac{1}{sin ( t )} d t = ln tan (\frac{t}{2})

\int sin (t) d t = - cos (t)

= ln tan (\frac{t}{2}) - (- cos (t))

Simplify

= ln tan (\frac{t}{2}) + cos (t)

Add a constant to the solution

= ln tan (\frac{t}{2}) + cos (t) + C

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