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Integration of Sec x Tan x
Integration of sec x tan x is the process of determining the integral of sec x tan x using different methods. As we know that integration is nothing but the reverse process of differentiation and the derivative of sec x is sec x tan x. So, using these facts we know that the integration of sec x tan x is equal to sec x + C, where C is the constant of integration. We can also determine the integration of sec x tan x using normal formulas of integration.
Let us determine the integral of sec x tan x using different methods and explore its formula and proof. Also, we evaluate the definite integration of sec x tan x from 0 to pi.
1.  What is Integration of Sec x Tan x? 
2.  Integration of Sec x Tan x Formula 
3.  Proof of Integration of Sec x Tan x 
4.  Integration of Sec x Tan x From 0 to Pi 
5.  FAQs on Integration of Sec x Tan x 
What is Integration of Sec x Tan x?
The integration of sec x tan x is sec x + C, where C is the integration constant. We know that the derivative of the trigonometric function sec x is sec x tan x, i.e., d(sec x)/dx = sec x tan x. So, if we integrate both sides of this equation we have ∫d(sec x)/dx dx = ∫sec x tan x dx ⇒ sec x + C = ∫sec x tan x dx. Hence, the integration of sec x tan x is mathematically, written as ∫sec x tan x dx = sec x + C. Integral of sec x tan x is nothing but its antiderivative only. Now, let us see the formula for the integration of sec x tan x.
Integration of Sec x Tan x Formula
The integral of sec x tan x is calculated using the derivative of sec x. The formula for the integration of sec x tan x is given by ∫sec x tan x dx = sec x + C, where C is the constant of integration.
Proof of Integration of Sec x Tan x
We know that the integration of sec x tan x can be done as the reverse process of the differentiation of sec x. Now, we will prove that the integral of sec x tan x is equal to sec x + C by writing sec x and tan x in terms of sin x and cos x. We will use the different formulas of trigonometry and the substitution method of integration. We can write sec x = 1/cos x and tan x = sin x/cos x. We have,
∫sec x tan x dx = ∫(1/cos x)(sin x/cos x) dx
= ∫(sin x/cos^{2}x) dx
Now, assume cos x = u. Differentiating both sides we have sin x dx = du ⇒ sin x dx = du [Because derivative of cos x is  sin x] Substituting these values in ∫(sin x/cos^{2}x) dx, we have
∫sec x tan x dx = ∫(1/u^{2}) du
= (u^{1}/(1)) + C
= (1/u) + C
= 1/cos x + C
= sec x + C
Hence we have proved that the integration of sec x tan x is equal to sec x + C.
Integration of Sec x Tan x From 0 to Pi
Now that we know that the integral of sec x tan x is sec x + C, we will calculate the definite integration of sec x tan x with limits from 0 to pi. We know sec 0 = 1 and sec π = 1. Therefore, we have
\(\begin{align} \int_{0}^{\pi} \sec x \tan x dx &=\left [ \sec x + C \right ]_0^{\pi}\\&=(\sec \pi + C) (\sec 0+C)\\&=(1+C)(1+C)\\&=1+C1C\\&=2\end{align}\)
Hence the definite integral of sec x tanx from 0 to pi is equal to 2.
Important Notes on Integration of Sec x Tan x
 The integration of sec x tan x is sec x + C, where C is the integration constant.
 Derivative of the trigonometric function sec x is sec x tan x implies that the integral of sec x tan x is sec x + C
Related Topics on Integration of Sec x Tan x
Integration of Sec x Tan x Examples

Example 1: Evaluate the integral of sec x tan x + sec^{2}x.
Solution: We know that the integration of sec x tan x is sec x + C and the integral of sec^{2}x is tan x + C.
Also, the integral of a sum of two functions is equal to the sum of integrals of the two functions. Therefore, we have
∫(sec x tan x + sec^{2}x) dx = ∫sec x tan x dx + ∫sec^{2}x dx
= sec x + tan x + C
Answer: Integral of sec x tan x + sec^{2}x is sec x + tan x + C

Example 2: Determine the integration of sec x + tan x.
Solution: We know that the integral of sec x is ln sec x + tan x + C and the integral of tan x is ln cos x + C.
So, the integral of sec x + tan x is,
∫(sec x + tan x) dx = ∫sec x dx + ∫tan x dx
= ln sec x + tan x + ln cos x + C
Answer: The integration of sec x + tan x is ln sec x + tan x + ln cos x + C.
FAQs on Integration of Sec x Tan x
What is Integration of Sec x Tan x in Calculus?
Integration of sec x tan x is the process of determining the integral of sec x tan x using different methods. It is given by sec x + C, where C is the integration constant.
What is the Formula for Integral of Sec x Tan x?
The integral of sec x tan x is calculated using the derivative of sec x. The formula for the integration of sec x tan x is given by ∫sec x tan x dx = sec x + C, where C is the constant of integration.
How to Calculate the Integration of Sec x Tan x?
The integral of sec x tan x can be calculated using the derivative of sec x. It can also be calculated by writing sec x and tan x in terms of sin x and cos x.
Is the Integration of Sec x Tan x the Same as the AntiDerivative of Sec x Tan x?
Yes, the integration of sec x tan x is the same as the antiderivative of sec x tan x as integration is the reverse process of differentiation.
What is the Antiderivative of Sec x Tan x?
The antiderivative of sec x tan x is equal to sec x + C, where C is the constant of integration.
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