What is the integral of cot(4t)sqrt(1-sin^2(4t)) ?

The integral of cot(4t)sqrt(1-sin^2(4t)) is 1/4 cot(4t)sin(4t)+1/4 ln|tan(2t)|+C

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integral of cot(4t)sqrt(1-sin^2(4t))

\int cot (4 t) 1 - sin^{2} (4 t) d t

\frac{1}{4} cot (4 t) sin (4 t) + \frac{1}{4} ln ∣ tan (2 t) ∣ + C

Solution steps

\int cot (4 t) 1 - sin^{2} (4 t) d t

Apply Integration By Parts

: \frac{1}{4} cot (4 t) sin (4 t) - \int - csc^{2} (4 t) sin (4 t) d t

= \frac{1}{4} cot (4 t) sin (4 t) - \int - csc^{2} (4 t) sin (4 t) d t

\int - csc^{2} (4 t) sin (4 t) d t = - \frac{1}{4} ln ∣ tan (2 t) ∣

= \frac{1}{4} cot (4 t) sin (4 t) - (- \frac{1}{4} ln ∣ tan (2 t) ∣)

Simplify

= \frac{1}{4} cot (4 t) sin (4 t) + \frac{1}{4} ln ∣ tan (2 t) ∣

Add a constant to the solution

= \frac{1}{4} cot (4 t) sin (4 t) + \frac{1}{4} ln ∣ tan (2 t) ∣ + C

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What is the integral of cot(4t)sqrt(1-sin^2(4t)) ?
The integral of cot(4t)sqrt(1-sin^2(4t)) is 1/4 cot(4t)sin(4t)+1/4 ln|tan(2t)|+C