What is the integral of csc(x)-sin(x) ?

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

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integral of csc(x)-sin(x)

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

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

Solution steps

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

\int csc (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

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