What is the integral of (csc(x)+tan(x))^2 ?

The integral of (csc(x)+tan(x))^2 is -cot(x)+2ln|tan(x)+sec(x)|-x+tan(x)+C

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integral of (csc(x)+tan(x))^2

\int (csc (x) + tan (x))^{2} d x

- cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x) + C

Solution steps

\int (csc (x) + tan (x))^{2} d x

Expand

(csc (x) + tan (x))^{2} : csc^{2} (x) + 2 csc (x) tan (x) + tan^{2} (x)

= \int csc^{2} (x) + 2 csc (x) tan (x) + tan^{2} (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^{2} (x) d x + \int 2 csc (x) tan (x) d x + \int tan^{2} (x) d x

\int csc^{2} (x) d x = - cot (x)

\int 2 csc (x) tan (x) d x = 2 ln ∣ tan (x) + sec (x) ∣

\int tan^{2} (x) d x = - x + tan (x)

= - cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x)

Add a constant to the solution

= - cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x) + C

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What is the integral of (csc(x)+tan(x))^2 ?
The integral of (csc(x)+tan(x))^2 is -cot(x)+2ln|tan(x)+sec(x)|-x+tan(x)+C