# Integration of tanx: Formula, Proof | tanx Integration

The integration of tanx is -ln|cosx| or ln|secx|, where ln denotes the natural logarithm, that is, logarithm with base e. Here we will learn how to find the integral of tanx dx.

## tanx Integration Formula

The tanx integration formula is given below.

• ∫tanx dx = -ln|cosx|+C
• ∫tanx dx = ln|secx|+C.

## Integration of tanx Proof

We will show that ∫tanx dx = -ln|cosx|+C. As $\tan x =\dfrac{\sin x}{\cos x}$, the integral of tanx will be equal to

∫tanx dx = $\int \dfrac{\sin x}{\cos x} \ dx$

Let us put cosx=z.

Differentiating, -sinx dx = dz.

So we have from above that

∫tanx dx = $\int \dfrac{-dz}{z}$

= – $\int \dfrac{dz}{z}$

= – ln|z|+C where C is a constant of integration

= -ln|cosx|+C as z=cosx.

So the integration of tanx is equal to -ln|cosx|+C (or) ln|secx|+C as secx=/cosx. This is obtained by the substitution method of integrations.

Video Solution on Integration of tanx:

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Integration cube root of x

Integration of 1/√x

Derivative of √x

## FAQs

Q1: What is the Integration of tanx?

Answer: The integration of tanx is -ln|cosx|+C where C is the integration constants.