∫1 / 1 − sinx dx

To solve the integral ∫ 1 / (1 - sin x) dx, we can use a clever substitution to simplify the expression. Here’s a step-by-step approach:

Step 1: Multiply by the Conjugate

First, we can multiply the numerator and the denominator by the conjugate of the denominator:

Conjugate: 1 + sin x

This gives us:

∫ (1 + sin x) / ((1 - sin x)(1 + sin x)) dx = ∫ (1 + sin x) / (1 - sin² x) dx

Step 2: Simplify the Denominator

Using the Pythagorean identity, we know that:

1 - sin² x = cos² x

So the integral becomes:

∫ (1 + sin x) / cos² x dx

Step 3: Split the Integral

We can split this into two separate integrals:

• ∫ (1 / cos² x) dx + ∫ (sin x / cos² x) dx

Step 4: Solve Each Integral

The first integral, ∫ (1 / cos² x) dx, is known to be:

tan x + C

The second integral, ∫ (sin x / cos² x) dx, can be solved using the substitution u = cos x, which gives:

-1/u = -tan x + C

Final Result

Combining both results, we have:

∫ 1 / (1 - sin x) dx = tan x - ln |cos x| + C

Thus, the solution to the integral is:

tan x - ln |cos x| + C