Problem 62 - Thomas Calculus

Question

The substitution \(u=\tan x\) gives $$ \int \sec ^{2} x \tan x d x=\int u d u=\frac{u^{2}}{2}+C=\frac{\tan ^{2} x}{2}+C $$ The substitution \(u=\sec x\) gives $$ \int \sec ^{2} x \tan x d x=\int u d u=\frac{u^{2}}{2}+C=\frac{\sec ^{2} x}{2}+C $$ Can both integrations be correct? Give reasons for your answer.

Step-by-Step Solution

Answer

No, both integrations are not correct due to inconsistency between results.

Step 1: Understand the Problem

We are given two different substitutions to integrate \( \int \sec^2 x \tan x \, dx \). We need to determine whether both methods are valid and if they provide consistent results when computing the integrals.

Step 2: Using the First Substitution (\(u = \tan x\))

For the first substitution \(u = \tan x\), we have \(du = \sec^2 x \, dx\). Thus, the integral becomes \(\int \sec^2 x \tan x \, dx = \int u \, du\). Evaluating this gives \(\frac{u^2}{2} + C = \frac{\tan^2 x}{2} + C\).

Step 3: Using the Second Substitution (\(u = \sec x\))

With the second substitution \(u = \sec x\), \(du = \sec x \tan x \, dx\). By rewriting the integral with this substitution: \(\int \sec^2 x \tan x \, dx = \int \sec x \cdot u \, du\) does not immediately simplify correctly without additional manipulation or presence of an error, but instead is mistakenly treated as \( \int u \, du \) leading to \( \frac{u^2}{2} + C = \frac{\sec^2 x}{2} + C\).

Step 4: Evaluate Consistency

The integrals \(\frac{\tan^2 x}{2} + C\) and \(\frac{\sec^2 x}{2} + C\) are not equivalent because \(\tan^2 x\) and \(\sec^2 x\) do not simplify to each other or differ only by a constant. Therefore, these results are inconsistent.

Step 5: Conclusion

Both attempts at substitution should yield equivalent antiderivatives for they are solving the same integral. The disparity suggests an error in understanding or execution of the second substitution, thus they cannot both be correct.

Key Concepts

Substitution Method

The substitution method is a crucial technique in integration used to simplify integrals by changing variables. The aim is to reduce a complex integral into a more manageable form.

Here's how it works:

• Choose a substitution for a part of the integral. This new variable replaces a function or part of the function, making the expression easier to integrate.
• Differentiate the substitution to find the differential, allowing you to express the integral entirely in terms of the new variable.
• Substitute back to the original variable after integrating, if necessary, to express your final answer using the original terms.

In the given problem, we used substitutions such as \(u = \tan x\) and \(u = \sec x\). However, it's crucial to remember that each substitution must lead to a correct and equivalent expression for the integral. Hence, checking the validity of results is key in using the substitution method effectively.

Trigonometric Integrals

Trigonometric integrals involve trigonometric functions like sine, cosine, tangent, and their reciprocal functions. They challenge problem-solvers to apply various techniques, such as trigonometric identities and substitutions.

When evaluating trigonometric integrals, consider:

• The form of the function: Is it in a format that simplifies using trigonometric identities or substitutions?
• Which substitution makes the integral easier? For example, letting \(u = \tan x\) or \(u = \sec x\) can simplify certain integrals involving \(\sec^2 x \tan x\).

Additionally, understanding these functions' derivatives and antiderivatives allows for substitutions that naturally arise from these relationships—like how the derivative of \(\tan x\) is \(\sec^2 x\). Using these relationships effectively can simplify solving integrals involving trigonometric functions.

Antiderivatives

Antiderivatives, or indefinite integrals, are functions whose derivative equals the given function. Finding antiderivatives requires reversing differentiation, often involving integration techniques.

When calculating antiderivatives with substitutions:

• Properly identify the substitution so that the integral becomes a simple, well-known form.
• After integration with the new variable, revert to the original variable setup to find the final antiderivative.
• Ensure consistency by checking if the antiderivative you've found satisfies the original integral condition.

In the exercise, substitutions involving \(u = \tan x\) and \(u = \sec x\) were explored. The former worked correctly, providing \(\frac{\tan^2 x}{2} + C\), while the latter incorrectly led to \(\frac{\sec^2 x}{2} + C\), revealing a mistake.

Calculus Problem-Solving

Successful calculus problem-solving involves choosing the right strategy among many possible techniques. One common approach is substitution, especially useful for complex integrals.

Steps to solve a calculus problem with integration may include:

• Understand the problem and identify suitable substitution possibilities.
• Check the consistency of your substitution by differentiating back to see if it equates to the original integrand.
• Verify your solution graphically or by differentiation to validate if your antiderivative is correct.

Missteps can occur, as seen with the incorrect attempt to substitute \(u = \sec x\) initially. The key is to persistently test and validate solutions for consistency, ensuring that the correct approach is used and that results are logically sound.