Question

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. $$ \int \tan ^{350} x \sec ^{2} x d x \text { or } \int \tan ^{350} x \sec x d x $$

Short answer

\(\int \tan^{350} x \sec x \, dx\) is more difficult due to less straightforward simplification possibilities.

Step-by-step solution

Analyze the Integral \(\int \tan^{350}x \sec^2 x \, dx\)

The integral \(\int \tan^{350} x \sec^2 x \, dx\) suggests a straightforward substitution because the derivative of \(\tan x\) is \(\sec^2 x\). This means we can set \(u = \tan x\), making \(du = \sec^2 x \, dx\). The integral simplifies to \(\int u^{350} \, du\), which is a basic power rule integral: \(\frac{u^{351}}{351} + C\). Substituting back, we get \(\frac{\tan^{351} x}{351} + C\), which is easy to evaluate.

Analyze the Integral \(\int \tan^{350}x \sec x \, dx\)

The integral \(\int \tan^{350} x \sec x \, dx\) is more challenging because it does not allow for easy substitution using \(\tan x\) or \(\sec x\). One may consider integration by parts or attempting to rewrite the functions in terms of sine and cosine, both of which complicate the integration process. There is no clear substitution or immediate simplification to a basic integral form.

Compare the Difficulties

\(\int \tan^{350} x \sec^2 x \, dx\) has a straightforward substitution due to the presence of \(\sec^2 x\), which directly corresponds to the derivative of \(\tan x\). In contrast, \(\int \tan^{350} x \sec x \, dx\) lacks such an immediate simplification and generally involves more complex methods such as integration by parts or trigonometric identities that do not simplify as directly. Thus, \(\int \tan^{350} x \sec x \, dx\) is more difficult.

Key concepts

Integration Techniques

Integration techniques are key tools in calculus for solving integrals, each suited to different kinds of functions. Recognizing which technique to use can greatly simplify the problem-solving process. These methods include:

• Substitution Method: Ideal for integrals that resemble a derivative chain rule.
• Integration by Parts: Suitable for products of functions where substitution is not straightforward.
• Partial Fractions and Trigonometric Identities: Useful for rational functions and trigonometric integrals.

Knowing various techniques broadens the range of integrals one can solve efficiently. When analyzing integrals, consider the form and elements present, such as derivatives or product types, to choose the best method.
Simplifying an integral early by identifying the appropriate technique saves time and reduces errors.

Substitution Method

The substitution method is an integration technique that transforms a complicated integral into a simpler one. The core idea is to change the variable of integration using a substitution that simplifies the integrand. Here's how it works:Choose a substitution: Look for a function and its derivative within the integrand. Set the substitution like this: if \ f(g(x))g'(x)dx \ is present, choose \(u = g(x)\) and then \(du = g'(x)dx\).Replace in the integral: Substitute \(u\) for \(g(x)\) and \(du\) for \(g'(x)dx\) to rewrite the integral in terms of \(u\).Integrate: Perform the integration on the new simpler form, which hopefully is a straightforward antiderivative.Substitute back: Replace \(u\) with the original function \(g(x)\) to return to the variable \(x\).Consider the example \ \int \tan^{350} x \sec^2 x \, dx \ used in the original exercise. Here, you can set \(u = \tan x\) and \(du = \sec^2 x \, dx\). The integral becomes \ \int u^{350} \, du \, a simple power integral. Using substitution is often a natural step when you recognize a function and its derivative appearing together.

Integration by Parts

Integration by Parts is a valuable technique for integrals involving products of functions where substitution method does not apply readily. Based on the product rule for differentiation, it is expressed as follows:\[\int u \, dv = uv - \int v \, du\]Where:

• Choose \(u\) as a function that becomes simpler when differentiated.
• Let \(dv\) be the remaining part of the integral, so that its antiderivative \(v\) is easy to find.

Let's look at \ \int \tan^{350} x \sec x \, dx \ from the exercise. Using integration by parts could be considered here, but it involves selecting functions \(u\) and \(dv\) carefully, which may lead to complex expressions. Often, rewriting using trigonometric identities or advanced algebra might be required prior to applying this technique.
Integration by Parts is powerful, especially for functions like products of polynomials and exponentials, or logarithms and trigonometric functions, where substitution might not strip the complexity sufficiently.