Evaluate ∫ 1 / (1 - tan x) dx

To evaluate the integral ∫ 1 / (1 - tan x) dx, we can use a substitution method. Let's start by rewriting the integrand.

Substitution

We can let:

• u = tan x

Then, the derivative of u with respect to x is:

• du/dx = sec² x

This implies:

• dx = du / sec² x

Since sec² x = 1 + tan² x, we can express sec² x in terms of u:

• sec² x = 1 + u²

Changing the Integral

Now, substituting into the integral gives:

∫ 1 / (1 - u) (du / (1 + u²))

This can be simplified to:

∫ du / [(1 - u)(1 + u²)]

Partial Fraction Decomposition

Next, we can use partial fractions to break this down:

1 / [(1 - u)(1 + u²)] = A / (1 - u) + (Bu + C) / (1 + u²)

By solving for A, B, and C, we can integrate each term separately.

Final Integration Steps

After finding the coefficients, we integrate each term:

• For A / (1 - u), the integral is -A ln|1 - u|.
• For (Bu + C) / (1 + u²), the integral is B (1/2) ln(1 + u²) + C arctan(u).

Back Substitution

Finally, substitute back u = tan x into the integrated result:

∫ 1 / (1 - tan x) dx = -A ln|1 - tan x| + B (1/2) ln(1 + tan² x) + C arctan(tan x) + C.

Thus, the integral is evaluated, and you can simplify further based on the values of A, B, and C obtained from the partial fraction decomposition.