Problem 23 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \sec^2 \theta \tan^3 \theta \, d\theta \)

Step-by-Step Solution

Answer

The integral is \(\frac{(\tan \theta)^4}{4} + C\).

Step 1: Identify u-substitution

To simplify the integration, use the substitution method. Notice that the derivative of \(\tan \theta \) is \(\sec^2 \theta\). Thus, let \(u = \tan \theta\).

Step 2: Compute du and substitution

Compute the differential \(du\). Since \(\frac{du}{d\theta} = \sec^2 \theta\), we have \(du = \sec^2 \theta \, d\theta\). Substitute in the integral: \(\int \tan^3 \theta \sec^2 \theta \, d\theta = \int u^3 \, du\).

Step 3: Integrate with respect to u

Now, integrate with respect to \(u\). We have \(\int u^3 \, du = \frac{u^4}{4} + C\), where \(C\) is the constant of integration.

Step 4: Substitute back to original variable

Substitute back \(u = \tan \theta\) to express the solution in terms of \(\theta\). The final expression is \(\frac{(\tan \theta)^4}{4} + C\).

Key Concepts

U-substitution

U-substitution is a powerful technique for solving integrals. It is essentially a method of simplifying the integral by making a clever change of variables.
This method is particularly useful when the integral contains a function and its derivative. Why? Because it conveniently transforms the integral into a more manageable form. Here's how it works:

• Identify a part of the integrand that can be set as a new variable, usually represented as "u".
• Calculate the differential of this new variable "u" (i.e., find what "du" is in terms of the original variable).
• Replace every occurrence of this function and its derivatives in the original integral with expressions involving "u" and "du".

The key is to choose "u" such that the rest of the integral becomes easier to integrate. In the context of trigonometric integrals, like the one given, recognizing a trigonometric identity derivative is often a great place to start. By setting "u" as \( \tan \theta \), which has a derivative \( \sec^2 \theta \), the integral simplifies beautifully.

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, tangent, and secant. They can often look intimidating, but with the right techniques, they become much more approachable.
Trigonometric integrals typically involve:

• Products or powers of trigonometric functions, such as \( \sin^m(x) \cos^n(x) \).
• Careful consideration of trigonometric identities and derivatives.

In the given problem, the function \( \sec^2 \theta an^3 \theta \) hints at \( \tan(\theta) \) having a straightforward derivative. Recognizing derivatives and utilizing identities like \( \sec^2(\theta) = 1 + \tan^2(\theta) \) can help solve these integrals.
This allows for the use of substitution to simplify the integral into a basic polynomial form, like moving from \( \int \sec^2 \theta \tan^3 \theta \, d\theta \) to \( \int u^3 \, du \). From there, it’s a simple power rule integration problem, which is much easier to solve.

Integration Techniques

Integration techniques are tactics used to simplify and solve integrals. They are not always straightforward and often involve picking the right method based on the type of integral you are facing. Here are a few common techniques:

• Substitution: As discussed, it is used when an integral appears to involve a function and its derivative.
• Integration by Parts: Useful for products of functions and is based on the product rule for differentiation.
• Partial Fractions: Useful for breaking down rational expressions into simpler terms.
• Trigonometric Identities: By leveraging identities such as \( \sin^2(x) + \cos^2(x) = 1 \), integrals become more approachable.

For the trigonometric integral in the exercise, u-substitution was ideal. By making \( u = \tan \theta \), it turned what initially appeared to be a complex integral into a straightforward problem.
Practice recognizing patterns and derivatives in integrals, as it often leads you to the right technique without struggle. When tackling advanced integrals, approaching them with adaptability and the right technique in mind can simplify the process significantly.