Solving the Integral of sin(2x)/[cos(4x)sin(4x)]

When faced with the challenge of solving the integral of ( frac{sin(2x)}{cos^4(x)sin^4(x)} )), one must apply a series of trigonometric identities and substitution techniques to simplify and eventually solve the problem.

Rewriting the Expression

The integral begins with:

[ int frac{sin(2x)}{cos^4(x) sin^4(x)} dx ]

Recalling the trigonometric identity for (sin(2x) 2sin(x)cos(x)), we can rewrite the integral as:

[ int frac{2sin(x)cos(x)}{cos^4(x) sin^4(x)} dx ]

Next, let's simplify the denominator using the identity (a^4 - b^4 a^2b^2 - 2a^2b^2), where (a sin(x)) and (b cos(x)).

[ cos^4(x) sin^4(x) (sin^2(x)cos^2(x))^2 - 2sin^2(x)cos^2(x) ]

Simplifying further, we get:

[ cos^4(x) sin^4(x) 1 - 2sin^2(x)cos^2(x) ]

Further Simplification with Substitution

Substituting (u sin(x)) and (du cos(x) dx), the integral transforms into:

[ int frac{2u}{1 - 2u^2(1 - u^2)} du int frac{2u}{1 - 2u^2 2u^4} du ]

To proceed further, we recognize that the integral is in a form that can be solved using a trigonometric identity or a standard table of integrals. Another approach could involve using a trigonometric substitution or numerical methods, depending on the context.

Alternative Solution

Another approach involves directly analyzing the structure of the integrand:

[ I int frac{sin(2x)}{sin^4(x) cos^4(x)} dx ]

This can be simplified as:

[ int frac{2sin(2x)}{(1 - cos^2(2x))^2} dx ]

Using the substitution (u cos(2x)), which gives (du -2sin(2x) dx), we substitute:

[ I -int frac{du}{1 - u^2} -arctan(u) C ]

Thus:

[ I -arctan(cos(2x)) C ]

Key Takeaways

The integral of (frac{sin(2x)}{cos^4(x) sin^4(x)} ) can be simplified and solved using trigonometric identities and substitution techniques. The key is recognizing the form of the integrand and applying the right substitutions to simplify the expression.

Keywords: integral, trigonometric integrals, calculus