What is the integral of (csc^2(4x))/(cot(4x)) ?

The integral of (csc^2(4x))/(cot(4x)) is -1/4 ln|cos(4x)|+1/4 ln|sin(4x)|+C

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integral of (csc^2(4x))/(cot(4x))

\int \frac{csc ^{2} ( 4 x )}{cot ( 4 x )} d x

- \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣ + C

Solution steps

\int \frac{csc ^{2} ( 4 x )}{cot ( 4 x )} d x

Rewrite using trig identities

= \int \frac{1 + cot ^{2} ( 4 x )}{cot ( 4 x )} d x

Expand

\frac{1 + cot ^{2} ( 4 x )}{cot ( 4 x )} : \frac{1}{cot ( 4 x )} + cot (4 x)

= \int \frac{1}{cot ( 4 x )} + cot (4 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 \frac{1}{cot ( 4 x )} d x + \int cot (4 x) d x

\int \frac{1}{cot ( 4 x )} d x = - \frac{1}{4} ln ∣ cos (4 x) ∣

\int cot (4 x) d x = \frac{1}{4} ln ∣ sin (4 x) ∣

= - \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣ + C

What is the integral of (csc^2(4x))/(cot(4x)) ?
The integral of (csc^2(4x))/(cot(4x)) is -1/4 ln|cos(4x)|+1/4 ln|sin(4x)|+C