Integration of tan x

The integration of tan x is an integral topic in calculus that deals with the finding of the indefinite integral of the tangent function. It is related closely to other integrals such as the integration of tan inverse x and the integration of log tan x but each differently. This tutorial discusses integration of tan x formula, derivation of integration of tan x formula, substitution method of integration, and provides solved examples for the integration of tan x for clarity.

 

Table Of Content

• What is the Integration of tan x?
• Integration of tan x Formula
• Integration of tan x Derivation
• Integration of Tan Inverse x
• Integration of Log tan x
• Examples on Integration of tan x
• Practice Problems on Integration of tan x
• Conclusion
• FAQs On Integration of tan x

 

What is the Integration of tan x?

The integration of tan x refers to the indefinite integral of the trigonometric function tan x with respect to x. That is:

∫ tan x dx

The integration of tan x gives a logarithmic expression in terms of the secant function. It is frequently requested in calculus and integral calculus lessons.

 

Integration of tan x Formula

∫ tan x dx = ln |sec x| + C

Where:

• ln is the natural logarithm

• |sec x| is the absolute value of sec x

• C is the constant of integration

This is the general integration of tan x formula applied to solve problems.

 

Integration of tan x Derivation

Let's derive the integration of tan x using the method of substitution of integration.

We know that:

 tan x = sin x / cos x

So,

 ∫ tan x dx = ∫ (sin x / cos x) dx

Let u = cos x

 Then, du = -sin x dx

∫ (sin x / cos x) dx = - ∫ (1/u) du = -ln |u| + C = -ln |cos x| + C = ln |sec x| + C

Hence,

 ∫ tan x dx = ln |sec x| + C

This completes the integration of tan x derivation using the substitution method of integration.

 

 Integration of Tan Inverse x

The integration of tan inverse x, or ∫ tan⁻¹x dx, varies from the integration of tan x. It employs integration by parts.

Formula:  
∫ tan⁻¹x dx = x · tan⁻¹x – (1/2) · ln(1 + x²) + C

Where:  

• C is the constant of integration  
• ln = natural logarithm  

Derivation (Step-by-step using integration by parts):  
Let’s apply the formula for integration by parts:  
∫ u · dv = u · v – ∫ v · du  

Choose:  
u = tan⁻¹x ⇒ du/dx = 1 / (1 + x²)  
dv = dx ⇒ v = x  

Now apply the integration by parts formula:  
∫ tan⁻¹x dx = x · tan⁻¹x – ∫ x · (1 / (1 + x²)) dx  

Now solve ∫ x / (1 + x²) dx:  
Let t = 1 + x² ⇒ dt = 2x dx  
So, x dx = (1/2) dt  

Then:  
∫ x / (1 + x²) dx = (1/2) ∫ dt / t = (1/2) ln|t| + C  
= (1/2) ln(1 + x²)  

Now substitute back:  

∫ tan⁻¹x dx = x · tan⁻¹x – (1/2) ln(1 + x²) + C

Final Answer:

∫ tan⁻¹x dx = x · tan⁻¹x – (1/2) · ln(1 + x²) + C

This is not to be confused with the integration of tan x, which is a trigonometric function only.

 

Integration of log tan x

The integration of log tan x, or ∫ log(tan x) dx, involves another method (generally integration by parts) and is more advanced.

It is not equal to the integration of tan x and is talked about in higher-level calculus problems.

Step-by-Step Explanation:

• We start with the integral:
  ∫ log(tan x) dx

• Use the logarithmic identity:
  log(tan x) = log(sin x) – log(cos x)

• Split the integral:
  ∫ log(tan x) dx = ∫ log(sin x) dx – ∫ log(cos x) dx

• Both ∫ log(sin x) dx and ∫ log(cos x) dx are complex and involve infinite series (Fourier series), so they are not typically solved directly at this level.

• However, we know a simplified and commonly accepted result:
  ∫ log(tan x) dx = log|tan x| + C

• This is the standard antiderivative used in calculus for ∫ log(tan x) dx.

• Final Answer:
  ∫ log(tan x) dx = log|tan x| + C, where C is the constant of integration.

 

Examples on Integration of tan x

Example 1:
Evaluate the integral:
∫ tan x dx

Solution:
We use the standard formula:
∫ tan x dx = log |sec x| + C

Answer:
log |sec x| + C

 

Example 2:
Find the value of ∫ tan(3x) dx

Solution:
Let’s use substitution:
Let u = 3x → du/dx = 3 → dx = du/3

∫ tan(3x) dx = ∫ tan(u) × (1/3) du
= (1/3) ∫ tan(u) du
= (1/3) log |sec u| + C
= (1/3) log |sec(3x)| + C

Answer:
(1/3) log |sec(3x)| + C

 

Example 3:
Evaluate ∫ tan x dx from x = π/4 to x = π/3

Solution:
∫ from π/4 to π/3 of tan x dx = [log |sec x|] from π/4 to π/3

Now calculate:
log |sec(π/3)| − log |sec(π/4)|
= log(2) − log(√2)
= log(2 / √2)
= log(√2)

Answer:
log(√2)

 

Example 4:
Evaluate: ∫ tan(2x + 1) dx

Solution:
Let u = 2x + 1 → du/dx = 2 → dx = du/2

∫ tan(2x + 1) dx = ∫ tan(u) × (1/2) du
= (1/2) ∫ tan(u) du
= (1/2) log |sec(u)| + C
= (1/2) log |sec(2x + 1)| + C

Answer:
(1/2) log |sec(2x + 1)| + C

These are basic examples on integration of tan x via direct use of the formula for integrating tan x.

 

Practice Problems - Integration of tan x

1. ∫ tan x dx

2. ∫ (tan x + tan⁻¹x) d

3. ∫ (sin x / cos x) dx

4. ∫ log(tan x) dx

5. ∫ from 0 to π/3 of tan x dx

These assist you in solidifying your grasp of the integration of tan x, the integration of tan inverse x, and integration of log tan x.

 

Conclusion

The integration of tan x is a basic outcome in integral calculus, and it comes to ln |sec x| + C. This integral is solved by the substitution method of integration and serves as an example of an indefinite integral. Although it appears to resemble other trigonometric integrals, the integration of tan x differs from that of the integration of tan inverse x and the integration of log tan x which need to be solved using different methods like integration by parts.

Mastering this subject entails learning the integration formula of tan x, exemplification with example problems on the integration of tan x, and application in different types of problems. Through frequent practice and clear understanding, you may be confident in solving problems based on the integration of tan x, the integration of tan inverse x, and the integration of log tan x.

 

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Frequently Asked Questions on Integration of tan x

1. What is the integral of tan x?

Ans: ∫ tan x dx = log |sec x| + C

• Where C is the integration constant.

 

2. What is the formula for tan x?

Ans: tan x = sin x / cos x

It is the sine to cosine ratio in trigonometry.

What is the definite integral of tan u?

Ans: ∫ tan u du = log |sec u| + C

Same form as tan x, only variable replaced from x to u.

 

3. What is the integral of tan u?

Ans: ∫ cot x dx = log |sin x| + C

 This is the standard integral of cotangent function.

 

4. What is the integration of cot x dx?

Ans: ∫ cot x dx = log |sin x| + C
This is the standard integral of cotangent function.

 

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