Integration of Tan X
The standard result of the integration of tan x is lnsec x + C. The trigonometric function tan x is integrable and this standard result of the integration of tan x is remembered as a formula. Let us learn how to solve the integration of tan x in the upcoming section.
1.  What is Integration of Tan X? 
2.  How to Solve Integration of Tan X? 
3.  Definite Integration of Tan X 
4.  Graph of Integration of Tan X 
5.  FAQs on Integration of Tan X 
What Is Integration of Tan X?
The integration of tan x is lncos x + C (or) lnsec x + C. The function f(x) = tan x is continuous at all real numbers, except x = (2n+ 1)π/2, The domain of the function = range of the function tan(x), except for the odd multiples of π/2. Hence tan x is integrable except for that interval with respect to x. We do the integration of tan x by the integration by substitution.
How to Solve Integration of Tan X?
To find the integration of tan x, with respect to x, we express tan x in terms of sine and cosine so that it becomes an integrable function. As per the definition of tan x, we have tan x = sin x / cos x
∫ tan x =∫ (sin x /cos x) .dx
This can be rewritten as \(\int \dfrac{1}{\cos x}\). sin x. dx
Let us find the indefinite integral of tan x using the substitution method of integration.
∫ f(g(x)) g'(x) dx = ∫ f(u) du = F(u) + C
Let u = cos x. Then du =  sin x . dx
⇒ dx =  du/ sin x
∫(sin x /cos x). dx =  ∫ du/ u
By the standard integration formula, we know that ∫ dx/x = ln x+ C
Thus ∫ (sin x /cos x) .dx =  ∫ du/ u =  lnu + c
= ln (cos x)+C
= ln (cos x) ^{1}+C
= ln (sec x) + C
∫ (sin x /cos x) .dx = ln (sec x) + C
∫ tan x = ln (sec x) + C
Thus the integration of tan x is lnsec x + C.
Definite Integration of Tan x
By the definition of the fundamental theorems of definite integrals, we can compute the definite integration of tan x between any two intervals. Let us compute the integration of tan x between π/6 and π/3.
We apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b)  f(a).
We know by the indefinite integration of tan x = lncos x + C. Here we take the absolute value only by computing the definite integrals.
Thus \(\int\limits_\dfrac{\pi }{3}^\dfrac{\pi }{6} tan(x) dx\) =
=lncos x\(^{\pi/2}_0\)
ln (cos \(\dfrac{\pi }{3}\))  ln (cos \(\dfrac{\pi }{6}\))
= ln ½  ln √3/2
Evaluating this further, we get lg √3 = ½ ln 3
Graph of Integration of Tan X
Let us evaluate the area under the graph tan x between 0 and π/2.
To find the \(\int\limits_0^\dfrac{\pi }{2}\)tan x dx, we apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b)  f(a).
\(\int\limits_0^\dfrac{\pi}{2}\)tan x dx
= lnsec x\(^{\pi/2}_0\)
= lnsec π/2  lnsec 0
=ln(∞) ln(1)
= ∞
Thus the graph of the integral of tan x diverges to infinity in the interval[0,π/2].
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Examples of Integration of Tan X

Example 1. Solve the integration of (tan x)^{2}
Solution:
To find the integral of (tan x)^{2}
(tan x)^{2 }= tan^{2} x
= sec^{2} x  1 (by the known trigonometric identity)
∫ tan^{2} x. dx= ∫ [sec^{2} x  1] dx
= ∫ (sec^{2} x). dx \(\int\) dx
Using the standard integration formula, we get
= tan x x + c
Answer: The integration of (tan x)^{2 }= tan x x + c

Example 2. What is the integration of tan (x/2) with respect to x?
Solution:
To find ∫ tan (x/2)
We know that integration of tan x = lncos x+ C
Applying, this we get ∫ tan (x/2) = 2 lncos x/2 + C
2 lncos x/2 = ln cos^{2} x/2 + C
= lnsec^{2 }x/2 + C
(or) = ln1 + tan^{2 }x/2 + C
Answer: The integration of tan (x/2) = lnsec^{2 }x/2 + C (or) ln1 + tan^{2 }x/2 + C

Example 3. Evaluate the integration of tan x in the interval 0 to π/4.
Solution:
Given f(x) = tan x
The integration of tan x = log sec x
To find the \(\int\limits_0^\dfrac{\pi }{4}\)tan x dx, we apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b)  f(a).
\(\int\limits_0^\dfrac{\pi}{4}\)tan x dx
= lnsec π/4  lnsec 0
=ln√2 ln1
= ln√2 0
= ln√2
= ln 2 ^{1/2} = ½ ln 2
Answer: The integration of tan x in the interval 0 to π/4 = ½ ln 2
FAQs on Integration of Tan X
What is Integration of Tan X?
The integration of tan x is lnsec x + C (or) lncos x + C.
Is Tan x Integrable?
Yes, Tan x is integrable. Tan x is a continuous function on its domain. The integration of tan x is lncos x + C.
How to do Integration of Tan X?
The integration of tan x is done by the method of integration by substitution. Tan x = sin x / cos x. Taking cos x as u, we get du = sin x dx. ∫ tan x = ∫ (sin x /cos x) .dx
=∫ du/ u = ln u + C
= lncos x + C.
Thus ∫ tan x = = lncos x + C.
What is Integration of 2Tan X?
The integration of tan x is lncos x + C. Thus \(\int\) 2 tan x = 2 \(\int\) tan x
∫ 2 tan x = 2 lncos x + C.
=  lncos^{2} x + C.
Is the Differentiation and Integration of Tan x the Same?
No. the differentiation and integration of tan x are not the same. The differentiation of tan x is sec^{2 }x and the integration of tan x is lnsec x + C.
What is The Technique We Use To Find The Integration of Tan X?
the integration of tan x is done by the method of integration by usubstitution. We write tan x in the integrable form sin x / cos x and then take u(x) is cos(x).
By the method of substitution, we know that ∫ f(g(x)) g'(x) dx =∫ f(u) du = F(u) + C, where g(x) = f(u). We apply this usubstitution technique for the integration of tan x and arrive at the standard result as ∫ tan x = log sec x
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