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.

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)

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