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Integration of Tan Square x
The formula for the integration of tan square x is tan x  x + C, with C as the integration constant. As we know that integration is nothing but the reverse process of differentiation, we can say that integration of tan square x is the same as the antiderivative of tan square x. We can determine the integral of tan^{2}x using the trigonometric identity or by writing tan x in terms of sin x and cos x. Mathematically, we write the integration of tan square x as ∫ tan^{2}x dx = tan x  x + C.
Let us calculate the integration of tan square x, determine its formula and the definite integral of tan square x with different limits. We will also solve some examples determining the integration of tan square x combined with some other trigonometric functions for a better understanding of the concept.
What is Integration of Tan Square x?
Integration of tan square x is the process of reverse differentiation of tan square x and finding its antiderivative. To find the integral of tan square x, we can use the trigonometric identities such as tan x = sin x/cos x and 1 + tan^{2}x = sec^{2}x. Mathematically, we can write the integration of tan square x as ∫ tan^{2}x dx = tan x  x + C, where ∫ is the symbol of integration and C is the integration constant. Let us now go through the formula of the integration of tan square x.
Integration of Tan Square x Formula
Now, let us understand the formula of integral of tan^{2}x. We can write the integration of tan square x mathematically as ∫ tan^{2}x dx = tan x  x + C, where C is the integration constant. We will understand how to get this formula further in this article by proving the integral of tan square x. The integration of tan square x is the tangent of angle x minus x plus the constant of integration which is given symbolically in the image below:
Integration of Tan Square x Proof
Now we know that the integration of tan square x is tan x  x + C which we will prove now using trigonometric identity sin^{2}x + cos^{2}x = 1. We will write tan x as sin x/ cos x. Therefore, we have
∫ tan^{2}x dx = ∫ (sin x/ cos x)^{2} dx
= ∫ (sin^{2}x / cos^{2}x) dx
= ∫ (1  cos^{2}x)/cos^{2}x dx [Because sin^{2}x + cos^{2}x = 1 ⇒ sin^{2}x = 1  cos^{2}x]
= ∫ (1/cos^{2}x) dx  ∫ 1 dx
= ∫ sec^{2}x dx  ∫ dx
= tan x  x + C [Because the derivative of tan x is sec^{2}x ⇒ integral of sec^{2}x is tan x + K]
We can also prove the integration of tan square x using the trigonometric identity 1 + tan^{2}x = sec^{2}x. To evaluate, we have
∫ tan^{2}x dx = ∫ (sec^{2}x  1) dx
= ∫ sec^{2}x dx  ∫1dx
= tan x  x + C
Hence, we have proved that the integration of tan square x is tan x  x + C.
Integration of Tan Square x From 0 to Pi by 4
We know that the formula for the integration of tan square x is tan x  x + C. Next, we will find its definite integral with limits from 0 to π/4 using this formula. For this, we have
\(\begin{align}\int_{0}^{\frac{\pi}{4}} \tan^2x \ dx&=\left [ \tan x  x + C \right ]_0^\frac{\pi}{4}\\&=\tan \frac{\pi}{4}  \frac{\pi}{4}+C  \tan 0+0C\\&=1\frac{\pi}{4}\end{align}\)
Hence the integration of tan square x from 0 to pi by 4 is equal to 1  π/4.
Important Notes on Integration of Tan Square
 The formula for the integration of tan square x is tan x  x + C.
 We can determine the integral of tan^{2}x using the trigonometric identities such as sin^{2}x + cos^{2}x = 1 and 1 + tan^{2}x = sec^{2}x.
 The integration of tan square x from 0 to pi by 4 is equal to 1  π/4.
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Integration of Tan Square x Examples

Example 1: Find the integration of tan square x sec square x.
Solution: We will find the integration of tan square x sec square x using the substitution method.
Assume u = tan x ⇒ du = sec^{2}x dx
∫tan^{2}x sec^{2}x dx = ∫u^{2} du
= u^{3}/3 + C
= (1/3) tan^{3}x + C
Answer: ∫tan^{2}x sec^{2}x dx = (1/3) tan^{3}x + C

Example 2: Evaluate the integration of tan square x minus cot square x.
Solution: To find the integration of tan square x minus cot square x, we will use trigonometric identities
 1 + tan^{2}x = sec^{2}x
 1 + cot^{2}x = cosec^{2}x
We have,
∫(tan^{2}x  cot^{2}x) dx = ∫[(sec^{2}x  1)  (cosec^{2}x  1)] dx
= ∫(sec^{2}x  1  cosec^{2}x + 1) dx
= ∫(sec^{2}x  cosec^{2}x) dx
= ∫sec^{2}x dx + ∫( cosec^{2}x) dx
= tan x + cot x + C [Because d(tan x)/dx = sec^{2}x and d(cot x)/dx =  cosec^{2}x]
Answer: Hence, ∫(tan^{2}x  cot^{2}x) dx = tan x + cot x + C
FAQs on Integration of Tan Square x
What is the Integration of Tan Square x in Calculus?
Integration of tan square x is the process of reverse differentiation of tan square x and finding its antiderivative. Mathematically, we can write the integration of tan square x as ∫ tan^{2}x dx = tan x  x + C.
What is the Formula for Integration of Tan Square x?
The formula for the integration of tan square x is tan x  x + C, with C as the integration constant.
How to Find Integration of Tan Square x?
We can determine the integral of tan^{2}x using the trigonometric identities such as sin^{2}x + cos^{2}x = 1 and 1 + tan^{2}x = sec^{2}x.
What is Integration of Tan Square x Sec Square x?
The integration of Tan Square x Sec Square x is given by, ∫tan^{2}x sec^{2}x dx = (1/3) tan^{3}x + C.
How to Find the Integration of Tan Square x Minus Cot Square x?
The Integration of Tan Square x Minus Cot Square x is written as ∫(tan^{2}x  cot^{2}x) dx = tan x + cot x + C.
What is the Integration of Tan Square x Minus Sin Square x?
The Integration of Tan Square x Minus Sin Square x is given by, ∫(tan^{2}x  sin^{2}x) dx = tan x  (3/2)x + (1/4) sin 2x + C.
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