# Integration of secx: Formula, Proof | Secx Integration

The integration of secx is equal to ln|secx+tanx| where ln denotes the natural logarithm, that is, the logarithm with base e. In this post, we will learn how to find the integral of secx dx.

## Integration of Secx Formula

The sec x integration formula is given below.

∫secx dx = ln|secx+tanx|+C

## Integration of Secx Proof

Let us show that ∫secx dx = ln|secx+tanx|+C. We have:

∫secx dx = ∫secx × 1 dx

= ∫secx $\times \dfrac{\sec x +\tan x}{\sec x + \tan x}$ dx

= ∫ $\dfrac{\sec^2 x +\sec x\tan x}{\sec x + \tan x}$ dx …(I)

Put secx+tanx = t.

Differentiating both sides, we have (sec2x+secxtanx)dx = dt.

Thus, we have from (I) that

∫secx dx = ∫ $\dfrac{dt}{t}$

= ln|t|+C where C is an integral constant, and ln stands for the natural logarithm, that is, the logarithm with base e.

= ln|secx+tanx|+C as t=secx+tanx.

So the integration of secx is ln|secx+tanx|+C where C is an integral constant.

Video Solution on Integration of secx:

Also Read:

Integration of tanx

Integration of cotx

Integration of rootx

## FAQs

Q1: What is the integration of secx?

Answer: The integration of secx is equal to ln|secx+tanx|+C.

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