How to Integrate ∫sec x tan x dx using Substitution: Step-by-Step Guide

∫sec x tan x dx

sec x + c

To integrate ∫sec x tan x dx, we’ll use a technique called substitution. Here is the step-by-step guide:

Step 1: Let u = sec x + tan x. Then, du/dx = sec x tan x + sec^2 x = (sec x)(tan x + sec x), which we can rewrite as:

du/dx = u(tan x + sec x) (since sec x + tan x = u)

Step 2: Rearrange the above equation to get:

1/(u(tan x + sec x)) du = dx

Step 3: Substitute the value of u and dx in the integral to get:

∫sec x tan x dx = ∫[1/(u(tan x + sec x))]du

Step 4: We can then integrate this expression with respect to u using the partial fraction decomposition. We can decompose the fraction as:

1/(u(tan x + sec x)) = A/(sec x) + B/(tan x)

where A and B are constants that we need to solve for.

Multiplying both sides by u(tan x + sec x), we get:

1 = A(tan x + sec x) + B(sec x)

Substituting x = 0 in the above equation, we get:

1 = A + B

Substituting x = π/4 in the above equation, we get:

1 = 2A

Solving for A and B, we get:

A = 1/2 and B = 1/2

Step 5: Substitute the values of A and B in the partial fraction decomposition to get:

1/(u(tan x + sec x)) = (1/2)/(sec x) + (1/2)/(tan x)

Step 6: Substitute this expression in the integral to get:

∫sec x tan x dx = ∫[(1/2)/(sec x) + (1/2)/(tan x)]du

Step 7: Integrating each term separately, we get:

∫(1/2)(sec x)du + ∫(1/2)(tan x)du

= (1/2)ln|sec x + tan x| + (1/2)ln|sec x| + C

where C is the constant of integration. Therefore, the final result is:

∫sec x tan x dx = (1/2)ln|sec x + tan x| + (1/2)ln|sec x| + C

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