What is the integral of sin(x)cot^2(x) ?

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

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integral of sin(x)cot^2(x)

\int sin (x) cot^{2} (x) d x

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

Solution steps

\int sin (x) cot^{2} (x) d x

Simplify

sin (x) cot^{2} (x) : cos (x) cot (x)

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

Rewrite using trig identities

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

Expand

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

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

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 ( x )} d x - \int sin (x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

\int a x^{- 1} d x

\int \frac{1}{( x + 1 ) ( x + 3 ) ^{2}} d x

\int \frac{2}{x ( x + 1 )} d x

\int sin (3 - 2 x) d x

\int \frac{x ^{3}}{1 + 3 1 + x ^{4}} d x

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