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Chapter 29: Problem 28

Evaluate \(\int \sec ^{3} x d x\) using a reduction formula.

Short Answer

Expert verified
The evaluation of the integral \(\int \sec ^{3} x dx\) requires a multi-step process involving the application of the reduction formula, followed by an integration by parts technique, and potentially additional steps to evaluate any remaining, more complex integral. While there isn't a simple answer without further given constraints, the key steps have been outlined in the provided solution.

Step by step solution

01

Apply the reduction formula

First, we are given that \(\int \sec ^{3} x d x\). We can rewrite \(\sec ^{3} x\) as \(\sec x . \sec ^{2} x\). Since we know that the derivative of \(\tan x\) is \(\sec^2x\), we can apply a reduction formula with \(u = \tan x\) and \(dv = \sec x dx\). We can then write our integral in the form of the integral of \(u.dv\).
02

Integration by parts

We can then use the method of integration by parts where \(鈭� udv = uv - 鈭� vdu\). Our \(u = \tan x\) and \(dv = \sec x dx\). We find \(du = \sec^2x dx\) and \(v = \ln |\sec x + \tan x|\). We plug these values into the formula for integration by parts.
03

Solve the integral

Computing the integral \(\int u dv = u.v - \int v du\) gives us \(\tan x \ln |\sec x + \tan x| - \int \ln |\sec x + \tan x| \sec^2 x dx\). The remaining integral can be more complex to evaluate and might involve additional reduction or trigonometric substitutions to solve.
04

Evaluate and simplify

Finally, the exercise involves the simplification of our resultant integral. We should simplify as much as possible, potentially using any additional substitution or integration techniques necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Parts
Integration by parts is a technique used to simplify integrals. Think of it as the reverse process of the product rule for differentiation. When you encounter a product of two functions within an integral, this method can be useful. The formula for integration by parts is given by:\[\int u \, dv = uv - \int v \, du\]Here, you need to choose which part of the integral will be \(u\) and which will be \(dv\). This choice can greatly affect how easy or difficult the integration process will be.
  • Begin by selecting \(u\), typically a part that becomes simpler when you differentiate.
  • Then find \(du\), the derivative of \(u\).
  • The remaining part is \(dv\), which you integrate to get \(v\).
Applying these steps leads to a new integral that might be simpler to solve. Be prepared to apply integration by parts more than once, especially if the new integral is still complex.
Reduction Formula
Reduction formulas are specialized tools in integration that simplify the process of integrating certain families of functions. These formulas relate complex integrals to simpler ones, reducing their difficulty.In our exercise, integrating \(\sec^3 x\) improves by recognizing it can be rewritten. By letting \(u = \tan x\) and \(dv = \sec x \, dx\), the integration by parts technique can be used effectively. This transforms the challenge into a form where the reduction formula connects higher powers of secant to lower ones.
  • Identify a repeated pattern within the integral.
  • Apply the reduction formula to step down the complexity.
  • Continue simplifying until reaching a basic, solvable integral.
Reduction formulas often require a clever combination of algebraic manipulation and trigonometric identities, probing deeper into simpler integral forms.
Trigonometric Integrals
Trigonometric integrals are those involving sine, cosine, tangent, and related functions. They demand a strong grasp of trigonometric identities and algebraic manipulations.In the integral \(\int \sec^3 x \, dx\), trig functions must be handled with care. Recognizing that the derivative of \(\tan x\) is \(\sec^2 x\) is crucial. This influences how you set up integration techniques like substitution or parts.
  • Reorganize the trigonometric integral into more manageable components.
  • Employ identities when necessary; for instance, \(\sec^2 x = 1 + \tan^2 x\).
  • This might lead to converting between trig and algebraic expressions for ease.
These integrals can sometimes require creative approaches rather than straightforward computation.
Definite Integrals
Definite integrals are those with specific limits of integration. Unlike indefinite integrals, which look for a general anti-derivative, definite integrals compute the net area under a curve between two points.While the exercise provided deals with an indefinite integral, understanding definite integrals aids in comprehension of boundary conditions and convergence.
  • After finding an antiderivative, apply the Fundamental Theorem of Calculus.
  • Evaluate the antiderivative at upper and lower bounds of the integral.
  • Subtract to find the total area: \[\int_a^b f(x) \, dx = F(b) - F(a) \]
Definite integral problems often lend themselves to real-world applications, where calculating total changes over interval matters.

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Most popular questions from this chapter

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