Что такое sec x
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Что такое sec x

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Integral of Sec x

To find the integral of sec x, we will have to use some facts from trigonometry. Sec x is the reciprocal of cos x and tan x can be written as (sin x)/(cos x). We can do the integration of secant x in multiple methods such as:

  • By using substitution method
  • By using partial fractions
  • By using trigonometric formulas
  • By using hyperbolic functions

We have multiple formulas for integration of sec x and let us derive each of them using the above mentioned methods. Also, we will solve some examples related to the integral of sec x.

1. What is the Integration of Sec x?
2. Integral of Sec x by Substitution Method
3. Integral of Sec x by Partial Fractions
4. Integral of Sec x by Trigonometric Formulas
5. Integral of Sec x by Hyperbolic Functions
6. FAQs on Integral of Sec x

What is the Integral of Sec x?

The integral of sec x is ln|sec x + tan x| + C. It denoted by ∫ sec x dx. This is also known as the antiderivative of sec x. We have multiple formulas for this. But the more popular formula is, ∫ sec x dx = ln |sec x + tan x| + C. Here «ln» stands for natural logarithm and ‘C’ is the integration constant. Multiple formulas for the integral of sec x are listed below:

  • ∫ sec x dx = ln |sec x + tan x| + C [OR]
  • ∫ sec x dx = (1/2) ln | (1 + sin x) / (1 — sin x) | + C [OR]
  • ∫ sec x dx = ln | tan [ (x/2) + (π/4) ] | + C
  • ∫ sec x dx = cosh -1 (sec x) + C (or) sinh -1 (tan x) + C (or) tanh -1 (sin x) + C

We use one of these formulas according to necessity.

Multiple formulas of integration of secant x are shown. Integral of sec x equals ln mod sec x plus tan x all plus c.

We will prove each of these formulas in different methods. Sounds interesting? Let’s go!

Integral of Sec x by Substitution Method

We can find the integral of sec x by substitution method. For this, we multiply and divide the integrand with (sec x + tan x). But why do we need to do this? Let us see.

∫ sec x dx = ∫ sec x · (sec x + tan x) / (sec x + tan x) dx

= ∫ (sec 2 x + sec x tan x) / (sec x + tan x) dx

Now assume that sec x + tan x = u.

Then (sec x tan x + sec 2 x) dx = du.

Substituting these values in the above integral,

∫ sec x dx = ∫ du / u = ln |u| + C

Substituting u = sec x + tan x back here,

∫ sec x dx = ln |sec x + tan x| + C

Integral of Sec x by Partial Fractions

To find the integration of sec x by partial fractions, we have to use the fact that sec x is the reciprocal of cos x. Wait! How can this be turned into partial fractions? Let us see.

∫ sec x dx = ∫ 1/(cos x) dx

Multiplying and dividing this by cos x,

∫ sec x dx = ∫ (cos x) / (cos 2 x) dx

∫ sec x dx = ∫ (cos x dx) / (1 — sin 2 x)

Now, assume that sin x = u. Then cos x dx = du. Then the above integral becomes

∫ sec x dx = ∫ du / (1 — u 2 )

By partial fractions, 1/(1 — u 2 ) = 1/2 [1/(1 + u) + 1/(1 — u)]. Then

∫ sec x dx = (1/2) ∫ [ 1/(1 + u) + 1/(1 — u) ] du

= (1/2) [ ln |1 + u| — ln |1 — u| ] + C

(We can get this by finding the integrals ∫ [ 1/(1 + u) ] du and ∫ [ 1/(1 — u) ] du separately by substitutiong 1 + u = p and 1 — u = q respectively).

By a property of logarithms, ln m — ln n = ln (m/n). So

∫ sec x dx = (1/2) ln | (1 + u) / (1 — u) | + C

Substituting u = sin x back here,

∫ sec x dx = (1/2) ln | (1 + sin x) / (1 — sin x) | + C

Integral of Sec x by Trigonometric Formulas

We can prove that the integral of sec x to be ln | tan [ (x/2) + (π/4) ] | + C by using trigonometric formulas. We can write sec x as 1/(cos x) where cos x can again be written as sin(x + π/2). So

∫ sec x dx = ∫ 1/(cos x) dx

Using half-angle formulas, sin A = 2 sin A/2 cos A/2. Applying this,

∫ sec x dx = ∫ 1 / [ 2 sin[ (x/2) + (π/4) ] cos[ (x/2) + (π/4) ] ] dx

= (1/2) ∫ 1 / [ sin[ (x/2) + (π/4) ] cos[ (x/2) + (π/4) ] ] dx

Multiplying and dividing the denominator by cos[ (x/2) + (π/4) ],

∫ sec x dx = (1/2) ∫ 1 / [ sin[ (x/2) + (π/4) ]/cos[ (x/2) + (π/4) ] · cos 2 [ (x/2) + (π/4) ] ] dx

= (1/2) ∫ sec 2 [ (x/2) + (π/4) ] / tan[ (x/2) + (π/4) ] dx

Now, assume that tan[(x/2) + (π/4)] = u. From this, (1/2) sec 2 [ (x/2) + (π/4) ] dx = du.

∫ sec x dx = ∫ du/u = ln |u| + C

Substituting u = tan[(x/2) + (π/4)] back,

∫ sec x dx = ln | tan [ (x/2) + (π/4) ] | + C

Integral of Sec x by Hyperbolic Functions

Using hyperbolic functions, we can prove that the integral of sin x to be tanh -1 (sin x) + C. For this, we use the substitution sec x = cosh t. Now, we have

tan x = √sec²x — 1 = √cosh²t — 1 = √sinh²t = sinh t

Differentiating both sides,

sec 2 x dx = cosh t dt

But sec x = cosh t.

(cosh 2 t) dx = cosh t dt

dx = (cosh t) / (cosh 2 t) dt = 1/(cosh t) dt

Substituting these values in ∫ sec x dx,

∫ sec x dx = ∫ (cosh t) [1/(cosh t) dt]

= cosh -1 (sec x) + C (This is because sec x = cosh t)

Similarly we can prove that ∫ sec x dx = sinh -1 (tan x) + C (or) ∫ sec x dx = tanh -1 (sin x) + C. Can you give them a try?

Therefore, ∫ sec x dx = cosh -1 (sec x) + C (or) sinh -1 (tan x) + C (or) tanh -1 (sin x) + C.

Important Notes Related to Integration of Sec x:

Here are the formulas of integral of secant x with the respective methods of proving them.

  • Using the substitution method,
    ∫ sec x dx = ln |sec x + tan x| + C
  • Using partial fractions,
    ∫ sec x dx = (1/2) ln | (1 + sin x) / (1 — sin x) | + C
  • Using trigonometric formulas,
    ∫ sec x dx = ln | tan [ (x/2) + (π/4) ] | + C
  • Using hyperbolic functions,
    ∫ sec x dx = cosh -1 (sec x) + C (or) sinh -1 (tan x) + C (or) tanh -1 (sin x) + C

Topics Related to Integral of Sec x:

Here are some topics that you may find helpful while doing the integration of sec x.

  • Integration Formulas
  • Integral Calculator
  • Derivative Formulas
  • Derivative Calculator

Solved Examples on Integration of Sec x

Example 1: Evaluate the definite integral ∫₀ π/2 sec x dx if it converges. Solution: The integral of sec x is, ∫ sec x dx = ∫ sec x dx = ln |sec x + tan x| + C ∫₀ π/2 sec x dx is obtained by applying the limits 0 and π/2 for this. Then ∫₀ π/2 sec x dx = [ ln |sec π/2 + tan π/2| + C ] — [ ln |sec 0 + tan 0| + C ] = Diverges This is because sec π/2 = ∞. Answer: ∫₀ π/2 sec x dx diverges.

Example 2: What is the value of ∫ (sec x + tan x) dx? Solution: We know that the integrals of secant x and tan x are ln |sec x + tan x| and ln |sec x| respectively. Thus, ∫ (sec x + tan x) dx = ∫ sec x dx + ∫ tan x dx = ln |sec x + tan x| + ln |sec x| + C Answer: ∫ (sec x + tan x) dx = ln |sec x + tan x| + ln |sec x| + C.

Example 3: Find the value of ∫ (sec 2 x + sec x) dx. Solution: ∫ (sec 2 x + sec x) dx = ∫ sec 2 x dx + ∫ sec x dx We know that ∫ sec 2 x dx = tan x and ∫ sec x dx = ln |sec x + tan x| + C. So ∫ (sec 2 x + sec x) dx = tan x + ln |sec x + tan x| + C Answer: ∫ (sec 2 x + sec x) dx = tan x + ln |sec x + tan x| + C.

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