Problem 10 - 5 Steps to A 5: AP Calculus AB 2021

Question

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \frac{1}{x^{2}} \sec ^{2}\left(\frac{1}{x}\right) d x$$

Step-by-Step Solution

Answer

Answer: The substitution used is \(u=\frac{1}{x}\).

Step 1: Perform substitution

Let's substitute \(u=\frac{1}{x}\).

Step 2: Find the derivative of u

Now, find the derivative of \(u\) with respect to \(x\): $$\frac{d u}{d x} = -\frac{1}{x^{2}}$$ From this, we can see that \(-x^{2} \frac{d u}{d x} = 1\), so \(d x = -\frac{1}{x^{2}} d u\).

Step 3: Substitute variables in the integral

Now, substitute the variables in the integral: $$\int \frac{1}{x^{2}} \sec ^{2}\left(\frac{1}{x}\right) d x = \int \sec ^{2}\left(u\right) (-\frac{1}{x^{2}} d u)$$

Step 4: Simplify the integral

The integral simplifies to: $$-\int \sec^{2}(u)d u$$

Step 5: Evaluate the integral

Now, evaluate the integral: $$-\int \sec^{2}(u)d u = -\tan(u) + C$$ where \(C\) is the constant of integration.

Step 6: Convert back to the original variable

Substitute our initial substitution back to convert the result to the variable \(x\): $$-\tan\left(\frac{1}{x}\right) + C$$ The final answer is: $$\boxed{-\tan\left(\frac{1}{x}\right) + C}$$

Key Concepts

Substitution Method

The substitution method is a powerful technique in integration that simplifies complex integrals by transforming them into a form that is easier to evaluate. This transformation is achieved by substituting a part of the integral with a single variable, typically denoted as \(u\).
By carefully choosing this substitution, you can streamline the integration process significantly.
In the given exercise, we start by setting \(u = \frac{1}{x}\).

• This choice usually arises when you see expressions that resemble derivatives or known integral forms.
• The next step is to find \( \frac{du}{dx} \), which helps in expressing \( dx \) in terms of \( du \). In this case, \( \frac{du}{dx} = -\frac{1}{x^2} \).
• This derivative allows us to substitute \(dx\) with \(-\frac{1}{x^2} du\).

These substitutions transform the initial integral into a simpler form, enabling the integration to proceed more smoothly.

Definite and Indefinite Integrals

Integrals are classified into two categories: definite and indefinite. Each serves a different purpose in calculus.

• Indefinite integrals represent a family of antiderivatives and include a constant of integration \(C\) because the process of differentiation eliminates this constant. The exercise involves evaluating an indefinite integral \(\int \sec^2(u) du\), resulting in \(-\tan(u) + C\).
• Definite integrals, on the other hand, have defined limits and compute the net area under the curve over a specified interval. While this exercise focuses on an indefinite integral, understanding both types is important for tackling various problems in calculus.

These integrals form the foundation for further calculus studies, including applications in physics and engineering.

Trigonometric Integrals

Trigonometric integrals involve integrals of trigonometric functions and require specialized techniques, such as the substitution method, to solve.
For instance, in this problem, we encounter the integral \(\int \sec^2(u) du\), a classic trigonometric integral that students often learn to evaluate as part of their calculus training.

• Recognizing standard forms like \(\int \sec^2(u) du = \tan(u) + C\) becomes handy as it directly provides solutions without requiring additional computation.
• Knowledge of these forms allows you to swiftly solve otherwise cumbersome integrals that frequently appear in calculus exercises such as these.

Being familiar with trigonometric identities and integrals is vital for success in calculus courses.

Calculus AB

Calculus AB is a course that covers differential and integral calculus' fundamental concepts, primarily focusing on simplifying methods to solve complex problems like the one in this exercise.
The curriculum generally includes essential calculus principles like:

• The limits and continuity.
• Derivatives and their applications.
• Integration techniques, including substitution and trigonometric integrals.

Understanding these topics prepares students for higher-level math courses and equips them with analytical skills applicable in various scientific fields. Within Calculus AB, mastering techniques such as the substitution method and recognizing standard integral forms, such as \(\sec^2(u)\), is crucial for achieving problem-solving proficiency and efficiency. Students, therefore, develop the ability to break down and tackle a range of calculus-related problems, similar to the given exercise.