The general form of an antiderivative of the function

f(x)= sec(x)(6 tan(x)−3 sec(x))
 
is    F(x)=?
  

 
 
Do not include the integration constant in the answer.

To find the antiderivative \( F(x) \) of the given function \( f(x) = \sec(x)(6 \tan(x) - 3 \sec(x)) \), we'll follow these steps:

### Step 1: Break down the function
The function can be written as:

\[
f(x) = 6 \sec(x) \tan(x) - 3 \sec^2(x)
\]

Now, we will find the antiderivative of each term separately.

### Step 2: Find the antiderivative of \( 6 \sec(x) \tan(x) \)
The antiderivative of \( \sec(x) \tan(x) \) is \( \sec(x) \). So, the antiderivative of \( 6 \sec(x) \tan(x) \) is:

\[
\int 6 \sec(x) \tan(x) \, dx = 6 \sec(x)
\]

### Step 3: Find the antiderivative of \( -3 \sec^2(x) \)
The antiderivative of \( \sec^2(x) \) is \( \tan(x) \). So, the antiderivative of \( -3 \sec^2(x) \) is:

\[
\int -3 \sec^2(x) \, dx = -3 \tan(x)
\]

### Step 4: Combine the results
Now, combining both terms, the general form of the antiderivative is:

\[
F(x) = 6 \sec(x) - 3 \tan(x)
\]

Thus, the antiderivative of \( f(x) = \sec(x)(6 \tan(x) - 3 \sec(x)) \) is:

\[
F(x) = 6 \sec(x) - 3 \tan(x)
\]

### Do you want more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What is the antiderivative of \( \sec^3(x) \)?
2. How do you find the antiderivative of a product of functions using integration by parts?
3. What is the relationship between the derivatives of trigonometric functions and their antiderivatives?
4. Can you explain how the chain rule applies in finding antiderivatives?
5. How does the antiderivative of \( \sec(x) \tan(x) \) compare to that of \( \csc(x) \cot(x) \)?

**Tip:** It's useful to memorize the antiderivatives of basic trigonometric functions like \( \sec(x) \tan(x) \) and \( \sec^2(x) \), as they often appear in calculus problems.

The general form of an antiderivative of the function f(x) = sec(x)(6 tan(x) − 3 sec(x)) is F(x) = ?

Explore the step-by-step solution for finding the antiderivative of the function f(x) = sec(x)(6 tan(x) − 3 sec(x)). This calculus problem covers the use of antiderivatives for trigonometric functions and applies the fundamental theorem of calculus. Suitable for advanced high school and undergraduate students.

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