What is the integral of (sec(2θ)-csc(θ/2)) ?

The integral of (sec(2θ)-csc(θ/2)) is 1/2 ln|tan(2θ)+sec(2θ)|-2ln|tan(θ/4)|+C

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integral of (sec(2θ)-csc(θ/2))

\int (sec (2 θ) - csc (\frac{θ}{2})) d θ

\frac{1}{2} ln ∣ tan (2 θ) + sec (2 θ) ∣ - 2 ln tan (\frac{θ}{4}) + C

Solution steps

\int sec (2 θ) - csc (\frac{θ}{2}) d θ

Apply the Sum Rule:

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

= \int sec (2 θ) d θ - \int csc (\frac{θ}{2}) d θ

\int sec (2 θ) d θ = \frac{1}{2} ln ∣ tan (2 θ) + sec (2 θ) ∣

\int csc (\frac{θ}{2}) d θ = 2 ln tan (\frac{θ}{4})

= \frac{1}{2} ln ∣ tan (2 θ) + sec (2 θ) ∣ - 2 ln tan (\frac{θ}{4})

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (2 θ) + sec (2 θ) ∣ - 2 ln tan (\frac{θ}{4}) + C

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What is the integral of (sec(2θ)-csc(θ/2)) ?
The integral of (sec(2θ)-csc(θ/2)) is 1/2 ln|tan(2θ)+sec(2θ)|-2ln|tan(θ/4)|+C