Indefinite Integral: Solving ∫(sec²x - 5)dx

Hey math enthusiasts! Today, we're diving into the world of calculus and tackling an indefinite integral. Specifically, we're going to determine the indefinite integral of the function ${\int\left(\sec ^2 x-5\right) d x}$. This involves finding a function whose derivative is ${\sec^2 x - 5}$. We'll also make sure to check our work using differentiation to ensure we've got the correct answer. Let's break it down step by step, making it easy to understand. We'll be using the basic rules of integration and some trigonometric identities to reach our solution. This is a fundamental concept in calculus and is essential for understanding more complex problems. Ready? Let's get started!

Understanding Indefinite Integrals and the Basics

First things first, what exactly is an indefinite integral? Think of it as the reverse process of differentiation. When we differentiate a function, we find its derivative, which represents the rate of change of the function. Integration, on the other hand, is the process of finding the antiderivative of a function. The antiderivative is a function whose derivative is equal to the original function. Since the derivative of a constant is zero, there are infinitely many antiderivatives for a given function, each differing by a constant. This is why we always include the constant of integration, usually denoted as '+ C', when finding indefinite integrals.

In our case, we're dealing with the integral of ${\sec^2 x - 5}$. The function ${\sec^2 x}$ is a trigonometric function, and 5 is a constant. We need to find a function whose derivative is ${\sec^2 x - 5}$. This involves applying the rules of integration to each term separately. The key here is recognizing the derivatives of basic trigonometric functions and the integration rules for constants and sums/differences of functions. Keep in mind that understanding the properties of integrals and their relationship with derivatives is crucial for mastering this concept. Remember, practice makes perfect! The more you work through these problems, the more comfortable you'll become. Let's delve into the specific steps to solve this integral and see how it all comes together.

Step-by-Step Solution of the Indefinite Integral

Now, let's solve the indefinite integral ${\int\left(\sec ^2 x-5\right) d x}$ step-by-step. We can break this integral into two separate integrals: ${\int \sec^2 x dx}$ and ${\int 5 dx}$. This is possible because of the linearity property of integrals, which states that the integral of a sum or difference of functions is the sum or difference of their integrals.

1. Integrating ${\sec^2 x}$: The integral of ${\sec^2 x}$ is a standard result. You should memorize that the derivative of ${\tan x}$ is ${\sec^2 x}$. Therefore, the integral of ${\sec^2 x}$ is ${\tan x}$. So, ${\int \sec^2 x dx = \tan x}$.
2. Integrating ${5}$: The integral of a constant, like 5, with respect to ${x}$ is simply the constant times ${x}$. Therefore, ${\int 5 dx = 5x}$.
3. Combining the Results: Now, we combine the results of both integrals. We have ${\int \sec^2 x dx = \tan x}$ and ${\int 5 dx = 5x}$. So, the integral of ${\sec^2 x - 5}$ is ${\tan x - 5x}$. Remember the constant of integration, ${C}$. Thus, ${\int \left(\sec ^2 x-5\right) d x = \tan x - 5x + C}$.

Therefore, the indefinite integral of ${\sec^2 x - 5}$ is ${\tan x - 5x + C}$. This is our final answer, but we're not done yet. We still need to check our work!

Checking the Solution through Differentiation

Verification is an essential step in solving integrals, as it helps confirm the accuracy of your answer. To verify our result, we'll differentiate the solution we obtained, ${\tan x - 5x + C}$, and see if we get back the original function, ${\sec^2 x - 5}$.

1. Differentiating ${\tan x}$: The derivative of ${\tan x}$ is ${\sec^2 x}$. This is a fundamental derivative you should have memorized. It's the foundation of this problem.
2. Differentiating ${-5x}$: The derivative of ${-5x}$ with respect to ${x}$ is ${-5}$. This follows the power rule of differentiation: the derivative of ${ax}$ is ${a}$.
3. Differentiating ${C}$: The derivative of a constant, ${C}$, is 0. Constants do not change with respect to ${x}$.
4. Combining the Derivatives: Combining the derivatives of each term, we get: ${\frac{d}{dx} (\tan x - 5x + C) = \sec^2 x - 5 + 0 = \sec^2 x - 5}$.

The derivative of our solution, ${\tan x - 5x + C}$, is indeed ${\sec^2 x - 5}$, which is the original function we started with. This confirms that our integration is correct! This process ensures that we understand the inverse relationship between differentiation and integration.

Tips and Tricks for Solving Indefinite Integrals

Here are some helpful tips and tricks to make solving indefinite integrals easier:

• Memorize Basic Integrals: Familiarize yourself with the integrals of common functions like trigonometric functions, exponentials, and polynomials. This will save you time and effort.
• Understand Integration Rules: Master the basic rules of integration, such as the power rule, the constant multiple rule, and the sum/difference rule. These rules form the basis of solving most integrals.
• Practice Regularly: The more you practice, the better you'll become at recognizing patterns and applying the appropriate techniques. Work through various examples to build your skills.
• Use Trigonometric Identities: Be familiar with trigonometric identities, as they can often simplify the integrand, making it easier to integrate. Identities such as ${\sin^2 x + \cos^2 x = 1}$ and double-angle formulas are particularly useful.
• Check Your Work: Always check your answer by differentiating it. This helps you identify errors and ensures your solution is correct. This step is crucial for reinforcing your understanding.
• Know Your Derivatives: Knowing the derivatives of common functions is critical. Integration is the reverse of differentiation, so you need to know both sides of the coin.
• Break Down Complex Integrals: If an integral looks complicated, try breaking it down into simpler parts using algebraic manipulation or trigonometric identities. This makes the integration process more manageable.

Conclusion: Mastering the Integral

Great job! We successfully determined the indefinite integral of ${\sec^2 x - 5}$ and verified our answer through differentiation. We've seen how understanding basic integration rules and trigonometric functions can help us solve these types of problems. Remember to always include the constant of integration, ${C}$, when finding indefinite integrals. Keep practicing, and you'll become more confident in your calculus skills.

Remember to review the fundamental concepts and rules, such as the derivatives of trigonometric functions and the linearity property of integrals. This will strengthen your foundation and improve your problem-solving abilities. Continuous practice and a solid understanding of the underlying principles are key to success in calculus. Keep learning, keep exploring, and enjoy the journey of mastering calculus!

For further learning, I highly recommend checking out resources on Khan Academy for more practice problems and detailed explanations. They offer a comprehensive collection of videos and exercises that can enhance your understanding of calculus. Good luck, and happy integrating!