What is the integral of x/(sin^2(x)cos^2(x)) ?

The integral of x/(sin^2(x)cos^2(x)) is -2xcot(2x)+ln|sin(2x)|+C

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integral of x/(sin^2(x)cos^2(x))

\int \frac{x}{sin ^{2} ( x ) cos ^{2} ( x )} d x

- 2 x cot (2 x) + ln ∣ sin (2 x) ∣ + C

Solution steps

\int \frac{x}{sin ^{2} ( x ) cos ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{8 x}{1 - cos ( 4 x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int \frac{x}{1 - cos ( 4 x )} d x

Apply Integration By Parts

: - \frac{1}{4} x cot (2 x) - \int - \frac{1}{4} cot (2 x) d x

= 8 (- \frac{1}{4} x cot (2 x) - \int - \frac{1}{4} cot (2 x) d x)

\int - \frac{1}{4} cot (2 x) d x = - \frac{1}{8} ln ∣ sin (2 x) ∣

= 8 (- \frac{1}{4} x cot (2 x) - (- \frac{1}{8} ln ∣ sin (2 x) ∣))

Simplify

8 (- \frac{1}{4} x cot (2 x) - (- \frac{1}{8} ln ∣ sin (2 x) ∣)) : - 2 x cot (2 x) + ln ∣ sin (2 x) ∣

= - 2 x cot (2 x) + ln ∣ sin (2 x) ∣

Add a constant to the solution

= - 2 x cot (2 x) + ln ∣ sin (2 x) ∣ + C

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\int cos (5 x) cos (11 x) d x

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