Problem 26 - Essential Calculus

Question

Find the indefinite integral. $$ \int \sec t(\sec t+\tan t) d t $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \sec t(\sec t+\tan t) d t\) is \(\frac{1}{3} (\sec t)^3 + \frac{1}{2} (\sec t)^2 + C\)

Step 1: Identify a possible substitution

Noticing that the derivative of \(\sec t\) is \(\sec t \tan t\), it might simplify things to let \(u = \sec t\). This would mean our differential \(du\) would then be \(\sec t \tan t dt\).

Step 2: Rewrite the integral in terms of \(u\)

Substituting \(u = \sec t\) into the integral, this then becomes \(\int u (u+ du)\). This simplifies down to \(\int (u^2 du+u du)\), leading to \(\int u^2 du + \int u du\).

Step 3: Evaluate the integrals

Integrating these two separate integrals results in the expressions \(\frac{1}{3} u^3 + \frac{1}{2} u^2\).

Step 4: Substitute back for \(t\)

Finally, replace the \(u\) back into the equation with \(\sec t\), resulting in the final answer \(\frac{1}{3} (\sec t)^3 + \frac{1}{2} (\sec t)^2 + C\)

Key Concepts

Understanding Integration Techniques

Integration is like the opposite of differentiation. It helps us find the accumulation of quantities, which is especially useful when dealing with areas under curves. Indefinite integrals give us a general form of antiderivatives with a constant of integration, usually denoted by C.
There are several integration techniques to tackle different types of integrals:

• Basic antiderivatives: Understanding the basic functions and their antiderivatives is essential. For example, the integral of a constant, or power functions, has rules that can be applied directly.
• Substitution method: This technique is used when the integral contains a product of a function and its derivative or can be transformed into one. This makes it easier to simplify the integral, as we'll see in this exercise.
• Integration by parts: Useful for integrals that are products of functions which cannot be simplified by substitution. This method uses the product rule of differentiation in reverse.

Choosing the right technique can make solving an integral much easier. Always look for patterns and simplifications in the integral problem you're facing.

Exploring the Substitution Method

The substitution method is a powerful tool for simplifying complicated integrals. In essence, it involves changing variables to make the integral easier to evaluate. Let’s break down how this works in a simple, clear manner.

First, you select a substitution that simplifies the expression, like converting a tricky product into a basic form. When you choose a substitution, you also define a new variable, say \(u\), which simplifies the integral. The most straightforward way to decide on \(u\) is to look for an inner function whose derivative is present in the integral—this makes the process seamless since it transforms the integral into an easily manageable format.

For this exercise, noticing the derivative structure of \(\sec t\) helped in choosing \(u = \sec t\). The differential \(du\) thus nicely matched the form \(\sec t \tan t dt\), making the substitution very effective. After substitution, you integrate in terms of \(u\), which simplifies the problem. Once you're done with this step, don't forget to convert back into the original variable using your substitution to complete the problem.

Working with Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions like \(\sin\), \(\cos\), \(\tan\), and \(\sec\). These can be challenging due to their periodic nature and their derivatives.

When dealing with these integrals, certain strategies come in handy:

• Look for identities: Trigonometric identities can simplify expressions dramatically. They help in rewriting parts of the integral into forms more suitable for straightforward integration or substitution.
• Use substitution or reduction: As we saw, using \(\sec t\) and its derivative \(\sec t \tan t\) for substitution simplifies the integration.
• Separate complex expressions: Often, breaking down a complex trigonometric product into simpler parts can ease integration, as seen in the exercise when splitting \(\sec t(\sec t + \tan t)\).

It's all about breaking the problem into simpler parts and using known trigonometric properties and techniques to solve them efficiently. Remember, practice makes perfect, and understanding these strategies deepens your calculus knowledge.