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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