Integral of sin(x) cos(x) / (sin(x)^4 cos(x)^4) dx

To solve this integral, we can use some trigonometric identities and algebraic manipulation.

Step 1: Simplify the expression
Let's simplify the expression inside the integral first. We can use the identity sin^2(x) + cos^2(x) = 1 to replace sin^4(x) and cos^4(x).

sin^4(x) = (sin^2(x))^2 = (1 - cos^2(x))^2 = 1 - 2cos^2(x) + cos^4(x)

Similarly, cos^4(x) = (1 - sin^2(x))^2 = 1 - 2sin^2(x) + sin^4(x)

Now, the expression becomes:

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

Step 2: Partial fraction decomposition
To proceed with the integration, we need to decompose the expression into partial fractions.

Let's assume:

A / sin^2(x) + B / sin^4(x) + C / cos^2(x) + D / cos^4(x) = sin(x) cos(x) / [(1 - 2cos^2(x) + cos^4(x))(1 - 2sin^2(x) + sin^4(x))]

Now, multiply both sides by [(1 - 2cos^2(x) + cos^4(x))(1 - 2sin^2(x) + sin^4(x))] to get:

A(1 - 2sin^2(x) + sin^4(x)) + B(1 - 2cos^2(x) + cos^4(x)) + C(sin^2(x) + sin^4(x)) + D(cos^2(x) + cos^4(x)) = sin(x) cos(x)

Step 3: Solve for the coefficients
To find the values of A, B, C, and D, we can substitute specific values of x that make some terms cancel out.

Let's substitute x = 0, π/2, π, and 3π/2 to obtain a system of equations:

A = 0
C + D = 0
A + B = 0
C = 1/2

From the equations above, we can determine that A = 0, B = 0, C = 1/2, and D = -1/2.

Step 4: Simplify the integral
Now that we have the partial fraction decomposition, we can rewrite the integral as:

∫ (0/sin^2(x) + 0/sin^4(x) + 1/2cos^2(x) - 1/2cos^4(x)) dx

This simplifies to:

∫ (1/2cos^2(x) - 1/2cos^4(x)) dx

Step 5: Evaluate the integral
To evaluate this integral, we can use the power