Integration of xdx/(1 sin2x cos 2x)

To integrate the given expression, we can use the method of substitution. Let's break down the steps involved:

Step 1: Identifying the Integral
The integral we need to solve is ∫ (x / (1 - sin^2(x) * cos^2(x))) dx.

Step 2: Simplifying the Integral
We can simplify the denominator by using the identity sin^2(x) + cos^2(x) = 1. By rearranging the terms, we get 1 - sin^2(x) * cos^2(x) = cos^2(x) - sin^2(x) * cos^2(x).

Step 3: Applying Substitution
Let's substitute u = cos^2(x) - sin^2(x) * cos^2(x). This substitution helps us simplify the integral further.

Differentiating u with respect to x, we get du/dx = -2sin(x)cos(x) - 2sin^2(x)cos^3(x).

Rearranging the terms, we have du = (-2sin(x)cos(x) - 2sin^2(x)cos^3(x)) dx.

Now, we need to find dx in terms of du. Dividing both sides of the equation by (-2sin(x)cos(x) - 2sin^2(x)cos^3(x)), we get dx = du / (-2sin(x)cos(x) - 2sin^2(x)cos^3(x)).

Step 4: Substituting the Variables
Substituting u and dx in the integral, we have ∫ (x / (1 - sin^2(x) * cos^2(x))) dx = ∫ (x / u) (du / (-2sin(x)cos(x) - 2sin^2(x)cos^3(x))).

Simplifying the expression, we get ∫ (-x / (2u)) du.

Step 5: Evaluating the Integral
Integrating ∫ (-x / (2u)) du with respect to u, we get -1/2 ∫ (x/u) du.

Using the power rule of integration, we have -1/2 ∫ (x/u) du = -1/2 ln|u| + C, where C is the constant of integration.

Step 6: Substituting Back
Finally, substituting u back in terms of x, we have -1/2 ln|cos^2(x) - sin^2(x) * cos^2(x)| + C.

Thus, the integral of xdx/(1 - sin^2(x) * cos^2(x)) is equal to -1/2 ln|cos^2(x) - sin^2(x) * cos^2(x)| + C.