What is the integral of (sec(4x)-csc((2x)/3)) ?

The integral of (sec(4x)-csc((2x)/3)) is 1/4 ln|tan(4x)+sec(4x)|-3/2 ln|tan(x/3)|+C

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integral of (sec(4x)-csc((2x)/3))

\int (sec (4 x) - csc (\frac{2 x}{3})) d x

\frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{3}{2} ln tan (\frac{x}{3}) + C

Solution steps

\int sec (4 x) - csc (\frac{2 x}{3}) 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 sec (4 x) d x - \int csc (\frac{2 x}{3}) d x

\int sec (4 x) d x = \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣

\int csc (\frac{2 x}{3}) d x = \frac{3}{2} ln tan (\frac{x}{3})

= \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{3}{2} ln tan (\frac{x}{3})

Add a constant to the solution

= \frac{1}{4} ln ∣ tan (4 x) + sec (4 x) ∣ - \frac{3}{2} ln tan (\frac{x}{3}) + C

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What is the integral of (sec(4x)-csc((2x)/3)) ?
The integral of (sec(4x)-csc((2x)/3)) is 1/4 ln|tan(4x)+sec(4x)|-3/2 ln|tan(x/3)|+C