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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 anti-derivative 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.

integration of sec x tan x

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/cos²x) 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/cos²x) dx, we have

∫sec x tan x dx = ∫(-1/u²) 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

\(\begin{align} \int_{0}^{\pi} \sec x \tan x dx &=\left [ \sec x + C \right ]_0^{\pi}\\&=(\sec \pi + C)- (\sec 0+C)\\&=(-1+C)-(1+C)\\&=-1+C-1-C\\&=-2\end{align}\)

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 + sec²x.

Solution: We know that the integration of sec x tan x is sec x + C and the integral of sec²x 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 + sec²x) dx = ∫sec x tan x dx + ∫sec²x dx

= sec x + tan x + C

Answer: Integral of sec x tan x + sec²x 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.

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Integration of Sec x Tan x Questions

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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?

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 Anti-Derivative of Sec x Tan x?

Yes, the integration of sec x tan x is the same as the anti-derivative of sec x tan x as integration is the reverse process of differentiation.

What is the Anti-derivative of Sec x Tan x?

The anti-derivative of sec x tan x is equal to sec x + C, where C is the constant of integration.

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