# Integral & Antiderivative of tanx: Formula, Proof

The antiderivative of tanx is loge|secx|+C where C is a constant. In this post, let us learn how to find the antiderivative of tanx.

The (integral or) antiderivative formula of tanx is given by

∫tanx dx = ln|secx| + C,

where ln denotes the logarithm with base e and C is an integral constant.

## Find the Antiderivative of tanx

The antiderivative of tanx is a function of x whose derivative is tanx. Thus, the antiderivative of tanx is the integration of tanx, that is,

∫ tanx dx.

To find it, let us put

ln secx = t

Differentiating both sides with respect to x, we have

$\dfrac{1}{\sec x}$ secx tanx = $\dfrac{dt}{dx}$

⇒ tanx dx = dt

So we have:

∫tanx dx

= ∫ dt

= t + C

= ln secx + C

As ln secx is defined for positive values of secx, we deduce that

∫tanx dx = ln|secx|+C.

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

Verification:

Lets verify that the derivative of ln|secx|+C is the function tanx. The derivative of ln|secx|+C is equal to

d/dx(ln|secx|+C)

= tanx + 0

= tanx, hence verified.

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## FAQs

### Q1: What is the antiderivative of tanx?

Answer: The antiderivative of tan x is ln|secx|+C where C is a constant of integration.