Evaluate the following integrals:

• ∫ (sin⁶x + cos⁶x) / (sin²x cos²x) dx

To evaluate the integral ∫ (sin 6 x + cos 6 x) / (sin 2 x cos 2 x) dx, we can simplify the expression using trigonometric identities.

Breaking Down the Integral

First, we can rewrite the denominator:

• sin 2x = 2 sin x cos x
• Thus, sin 2x cos 2x = (1/2) sin 4x

Now, the integral becomes:

∫ (sin 6 x + cos 6 x) / ((1/2) sin 4 x) dx = 2 ∫ (sin 6 x + cos 6 x) / sin 4 x dx

Using Trigonometric Identities

Next, we can use the identities for sin and cos to express sin 6x and cos 6x in terms of sin 4x and cos 4x:

• sin 6x = sin(4x + 2x) = sin 4x cos 2x + cos 4x sin 2x
• cos 6x = cos(4x + 2x) = cos 4x cos 2x - sin 4x sin 2x

Substituting these into the integral gives:

2 ∫ [(sin 4x cos 2x + cos 4x sin 2x) + (cos 4x cos 2x - sin 4x sin 2x)] / sin 4x dx

Combining Terms

This simplifies to:

2 ∫ [(cos 2x + sin 2x) dx]

Final Integration

Now, we can integrate:

• ∫ cos 2x dx = (1/2) sin 2x
• ∫ sin 2x dx = -(1/2) cos 2x

Putting it all together, we have:

2 [(1/2) sin 2x - (1/2) cos 2x] + C

Result

The final answer is:

sin 2x - cos 2x + C