Evaluate: ∫ 1 / sin² x cos² x dx

To evaluate the integral ∫ 1 / sin² x cos² x dx, we can simplify the expression using trigonometric identities.

Using Trigonometric Identities

Recall that:

• sin² x = 1 - cos² x
• cos² x = 1 - sin² x

We can rewrite the integral as:

∫ csc² x sec² x dx

Substitution Method

Next, we can use the substitution:

Let u = tan x, then du = sec² x dx.

This transforms our integral into:

∫ csc² x du

Expressing csc² x in terms of u

Since csc² x = 1 + tan² x = 1 + u², we can rewrite the integral:

∫ (1 + u²) du

Integrating

Now, we can integrate:

• ∫ 1 du = u
• ∫ u² du = (u³)/3

Thus, the integral becomes:

u + (u³)/3 + C

Back Substitution

Substituting back for u, we have:

tan x + (tan³ x)/3 + C

Final Result

The evaluated integral is:

∫ 1 / sin² x cos² x dx = tan x + (tan³ x)/3 + C