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 (sec^{2}x+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.

Have You Read These Integrations?

Integration of tanx | Integration of cotx |

Integration of cosecx | Integration of root(x) |

Integration of 1/x | Integration of ln(x) |

Video Solution on Integration of secx:

## FAQs

**Q1: What is the integration of secx?**

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