What is the Integration of Sec x? (Formula & Proof)

You cannot solve this integral using normal, straightforward logic. To prove this formula, mathematicians had to invent a highly unnatural, massive 'Magic Trick' step.

Step 1: The Magic Multiplication We take the original problem ∫ sec(x) dx and forcefully multiply both the numerator and the denominator by the exact same complex block: [sec(x) + tan(x)]. So, the integral becomes: ∫ [ sec(x) * (sec(x) + tan(x)) ] / [ sec(x) + tan(x) ] dx

Step 2: Expand the Numerator Multiply the sec(x) inside the top bracket: ∫ [ sec²(x) + sec(x)tan(x) ] / [ sec(x) + tan(x) ] dx

Step 3: Substitute the Denominator Let the entire bottom denominator be 'u': u = sec(x) + tan(x)

Step 4: Differentiate 'u' Now, we find the derivative of 'u' with respect to x (du/dx):

• The derivative of sec(x) is sec(x)tan(x).
• The derivative of tan(x) is sec²(x). So, du = [sec(x)tan(x) + sec²(x)] dx.

Notice the massive miracle! The derivative (du) is exactly 100% identical to the massive numerator sitting at the top of our integral.

Step 5: Replace and Solve Because the entire messy top part is simply 'du', and the bottom part is 'u', our giant, terrifying integral completely collapses into the easiest integral in the world: ∫ (1/u) du

Step 6: The Logarithm We mathematically know that the integral of (1/u) is simply the natural log: ln|u| + C. Finally, substitute the original value of 'u' back into the answer to get the final formula: ln |sec(x) + tan(x)| + C.