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Integration of tanx | Formula | Solved Example |

The integration of tanx gives us ln|secx|+c. Furthermore, we will also see the derivation of the integral of tanx.

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Posted by Tauseef Arshad, 3/10/2021
Integration of tanx | Formula | Solved Example |

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The integration of tanx is given as:

Integration of tanx

The integration of tanx gives us ln|secx|+c. Furthermore, we will also see the derivation of the integral of tanx.


Integration of tanx – Modified Formula

We know, Integration of tanx is given as ln|secx|+c. For the ease of memorization we can modify it.
We know, secx = 1/cosx
By replacing secx we will get-
∫ tanx dx = ln|1/cosx| + c = ln|1| – ln|cosx| + c = 0 – ln|cosx| + c
Therefore, ∫ tanx dx = – ln|cosx| + c

Hence, ∫ tanx dx = – ln|cosx|+c (or) ln|secx|+c.


Derivation of Integration of tanx

Firstly, we need to express tanx in terms of sinx and cosx.

∫ tanx dx = ∫ sinx/cosx dx

Now, Let t = cos x. Then dt = – sin x . dx

⇒ dx = – dt/ sin x

∫(sin x /cos x). dx = – ∫ dt/ t

Also, we know that ∫ dt/t = ln t + c

Thus ∫ (sin x /cos x) .dx = – ∫ dt/ t = – ln|t| + c

⇒ -ln |(cos x)+c = ln |(cos x)-1|+c

= ln (sec x) + c

∫ (sin x /cos x) .dx = ln (sec x) + c

Therefore, ∫ tan x = ln|sec x| + c

Hence, the integration of tan x = ln|sec x| + c.


Solved Example for Reference

For Example : What is the value of the integral ∫tan (x/4) dx?

Solution: We know that integration of tan x = -ln|cos x|+ c

By applying this, we get ∫ tan (x/4) = -4 ln|cos x/4| + c

⇒ -4 ln|cosx /4| = -ln |cos⁴x /4| + c = ln |(cos⁴x /4)-1| + c

= ln|sec⁴x /4| + c

Hence, ∫tan (x/4) dx = ln|sec⁴x /4| + c


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