Integration of tan x

The integration of tan x is an important topic in calculus that focuses on finding the indefinite integral of the tangent function. In simple terms, it helps us reverse the process of differentiation for tan x. This topic is closely related to other integrals, such as the integration of tan⁻¹x and the integration of log(tan x), but each one requires a different method to solve.

In this guide, you will learn the standard formula for integrating tan x, understand its derivation using the substitution method, and see how it differs from the integration of tan inverse x and log tan x. Along with step-by-step explanations, the guide provides solved examples and practice problems to make the concept easier to understand.

 

Table Of Contents

• 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 means finding a function whose derivative is tanx. In calculus, this is written as∫tanxdx. The result of this integration is a logarithmic expression that involves the secant function. By using the substitution method, where we write tan 𝑥=sin𝑥cos𝑥, the integral simplifies and finally gives the resultln⁡|sec x|+C, where C is the constant of integration. In simple words, the integration of tan x leads to a logarithmic form, and the formula we use is:

∫tan𝑥 d𝑥=ln⁡|sec 𝑥|+C

This is the standard formula applied whenever we need to find the integral of the tangent function.

 

Integration of tan x Formula

∫tan𝑥 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

ln represents the natural logarithm, ∣sec𝑥∣ is the absolute value of the secant function, and 𝐶 is the constant of integration. This formula is very useful in solving calculus problems because it allows us to integrate the tangent function quickly without going through the full derivation each time. In simple words, whenever you see the integral of tan𝑥, you can directly apply this formula to get the answer.

 

Integration of tan x Derivation

The integration of tan x is one of the standard results in calculus. To derive it, we start by rewriting tan x in a more convenient form and then apply substitution.

Step 1: Express tan x in terms of sin and cos
We know:
tan𝑥=sin𝑥cos𝑥

So, the integral becomes:
∫tan𝑥 d𝑥=∫(sin𝑥cos𝑥)d𝑥

Here, expressing tan x as a ratio of sine and cosine helps us apply substitution easily.

 

Step 2: Use substitution
Let

u=cos𝑥

Then,dudx=−sin⁡x

or, du=–sin𝑥d𝑥

Now substitute these values into the integral:

∫(sin⁡xcos⁡x)dx=∫(−dudx)

 

Step 3: Solve the simplified integral

∫(−dudx)=(1u)du

=−ln⁡|u|+C

Here, C is the constant of integration.

 

Step 4: Back-substitute u = cos x
Since we initially set u = cos x, substitute it back:

∫tan 𝑥 d𝑥=–ln⁡|cos𝑥|+C

 

Step 5: Alternate form of the result
We know that

−ln⁡|cosx|ln⁡(1/|cosx|)=ln⁡|secx|

Therefore, the final result can be written in two equivalent ways:

∫tan 𝑥 d𝑥=–ln⁡|cos𝑥|+C

or

∫tan 𝑥 d𝑥=ln⁡|sec𝑥|+C

Final Answer
The integration of tan x is:
∫tan 𝑥 d𝑥=–ln⁡|sec𝑥|+C

 

Integration of Tan Inverse x

Inverse trigonometric functions are the reverse process of trigonometric functions. For example, while tan θ = x, the inverse function gives θ = tan⁻¹x. These functions are very useful in calculus, as they help us evaluate integrals that involve ratios and square roots. One important example is the integration of tan⁻¹x, which is commonly asked in exams and applied in solving higher-level calculus problems.

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

Formula:  
∫tan−1⁡x dx=x×tan−1−(12)×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−1⁡x⇒dudx=1(1+x2)

dv=dx⇒v=x

Now apply the integration by parts formula:  

∫tan−1⁡x dx=x·tan−1⁡x−∫x·(1(1+x2))dx

 

Now solve ∫x(1+x2)dx:  

Let t=1+x2⇒dt=2x dx

 

So,

$x \ dx = \left(\frac{1}{2} \right) dt  $

Then:  
∫x(1+x2)dx=∫dtt=(12)ln⁡|t|+C

=(12)ln⁡(1+x2)

 

Now substitute back:  

∫tan−1⁡x dx=x·tan−1⁡xln⁡(1+x2)+C

Final Answer:

∫tan−1⁡x dx=x·tan−1⁡x−(12)·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 the 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.

hrough 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.

 

Frequently Asked Questions on Integration of tan x

1. What is the integral of tan x?

Answer: ∫ tan x dx = log |sec x| + C

• Where C is the integration constant.

 

2. What is the formula for tan x?

Answer: 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?

Answer: ∫ cot x dx = log |sin x| + C

This is the standard integral of cotangent function.

 

4. What is the integration of cot x dx?

Answer: ∫ cot x dx = log |sin x| + C

This is the standard integral of cotangent function.

 

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