What is the integral of (1-csc(t)cot(x)) ?

The integral of (1-csc(t)cot(x)) is t-cot(x)ln|tan(t/2)|+C

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integral of (1-csc(t)cot(x))

\int (1 - csc (t) cot (x)) d t

t - cot (x) ln tan (\frac{t}{2}) + C

Solution steps

\int 1 - csc (t) cot (x) 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 1 d t - \int cot (x) csc (t) d t

\int 1 d t = t

\int cot (x) csc (t) d t = cot (x) ln tan (\frac{t}{2})

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

Add a constant to the solution

= t - cot (x) ln tan (\frac{t}{2}) + C

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What is the integral of (1-csc(t)cot(x)) ?
The integral of (1-csc(t)cot(x)) is t-cot(x)ln|tan(t/2)|+C