Problem 65 - Thomas Calculus

Question

Use reduction formulas to evaluate the integrals. \(\int 2 \sec ^{3} \pi x d x\)

Step-by-Step Solution

Answer

\(\frac{2}{\pi} \sec(\pi x) \tan(\pi x) + \frac{2}{\pi} \ln |\sec(\pi x) + \tan(\pi x)| + C\)

Step 1: Recognize the Integral Form

The integral \(\int 2 \sec^3 \pi x \, dx\) involves the function \(\sec^3 x\), which is known to use a reduction formula. Identify if there's a simpler form or starting point for the involved integral.

Step 2: Use the Reduction Formula for Secant Function

The reduction formula for \(\int \sec^n x \, dx\) is given by:\[\int \sec^n x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx\]For \(n = 3\), this becomes:\[\int \sec^3 x \, dx = \frac{\sec x \tan x}{2} + \frac{1}{2} \int \sec x \, dx\]

Step 3: Evaluate the Integral of \(\sec x\)

Recall that:\[\int \sec x \, dx = \ln |\sec x + \tan x| + C\]This will be substituted back into the expression found in Step 2.

Step 4: Substitute the Results Back

Substitute the result from Step 3 into the reduction formula for \(\int \sec^3 x \, dx\) found in Step 2:\[\int \sec^3 x \, dx = \frac{\sec x \tan x}{2} + \frac{1}{2} (\ln |\sec x + \tan x| + C)\]

Step 5: Adjust for the Original Integral

Remember the original integral was \(\int 2 \sec^3 \pi x \, dx\). Therefore, multiply the entire expression from Step 4 by 2:\[2 \left( \frac{\sec x \tan x}{2} + \frac{1}{2} \ln |\sec x + \tan x| \right)\]Simplify to get:\[\sec x \tan x + \ln |\sec x + \tan x| + C\]

Step 6: Handle the \(\pi x\) Alignment

Since we have the integral of \(\sec(\pi x)\), not just \(\sec x\), we replace \(x\) by \(\pi x\) in our final expression and account for the differentiation factor \(1/\pi\) due to substitution:\[\frac{1}{\pi} \left( \sec(\pi x) \tan(\pi x) + \ln |\sec(\pi x) + \tan(\pi x)| \right) + C\]

Step 7: Finalize the Expression

Combine all steps to write the complete evaluated integral expression:\[\int 2 \sec^3 \pi x \, dx = \frac{2}{\pi} \sec(\pi x) \tan(\pi x) + \frac{2}{\pi} \ln |\sec(\pi x) + \tan(\pi x)| + C\]

Key Concepts

Reduction Formulas

Reduction formulas are powerful tools in calculus, especially useful for handling complex integrals that involve powers of trigonometric functions. They work by reducing the integral of a function raised to a power, to an expression involving simpler integrals and the function raised to a lower power.

When you see an integral like \( \int \sec^n x \, dx \, \), a reduction formula can be your go-to strategy. You start with the standard reduction formula:

• \( \int \sec^n x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx \)

This formula allows you to break down the integral involving \(\sec^n x\) to a simpler integral form, making the problem easier to solve.

With reduction formulas, the key is to simplify a higher power of a function into lower ones, potentially linking it directly to a basic integral that you can solve more easily, like \(\int \sec x \, dx\). This method significantly reduces the algebraic complexity and aids in systematic evaluation.

Secant Function Integration

Integrating functions involving the secant (\(\sec x\)) function often requires specific techniques. The secant function itself is \(\sec x = \frac{1}{\cos x}\), which makes its integration non-trivial. Regular integration techniques might not directly apply, which is why reduction formulas become crucial.

When integrating \(\sec^3 x\), recognizing its special structure is key. You utilize a known formula that specifically addresses \(\int \sec^3 x\) to break it down accurately.

• First, apply the reduction formula to handle the secant raised to the third power.
• This rewrites the function in terms of \(\sec x\), which has a direct integration formula.

These types of problems benefit from knowing trigonometric identities and being comfortable manipulating these expressions, preparing them to be substituted back into the integral to find a solution.

Integral of Secant

The integral of the simple secant function, \(\int \sec x \, dx\), is one of those peculiar integrals that doesn’t fit into the basic set of formulas you may have learned in calculus classes. Yet, it is vital in solving more complex integrals involving secant functions.

• The integral is given by \(\int \sec x \, dx = \ln |\sec x + \tan x| + C\).
• This result comes from a clever substitution or using known identities, leading to a log expression which isn’t immediately obvious.

Understanding this integral is crucial because in more complex expressions like \(\int \sec^3 x \, dx\), once a reduction formula is applied, you'll end up needing \(\int \sec x \, dx\).

This role of \(\int \sec x \, dx\) as a simpler component of more complicated integrals highlights the importance of memorizing specific non-intuitive results in calculus, equipping you to tackle a broader range of problems efficiently.