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Chapter 7: Problem 8

$$\text { How would you evaluate } \int \sec ^{12} x \tan x d x ?$$

Short Answer

Expert verified
#Question# Evaluate the integral of sec^12(x) * tan(x) with respect to x. #Answer# The integral of sec^12(x) * tan(x) with respect to x is sec^12(x)ln|sec(x)| - 12(鈭玸ec^14(x)ln|sec(x)|dx - 鈭玸ec^12(x)ln|sec(x)|dx). However, the integral becomes quite complicated and may not have a simple closed-form solution. It is common to solve such integrals using numerical methods or approximations, depending on the purpose of the integral.

Step by step solution

01

Choose Function for Integration by Parts

In order to use integration by parts, we will first choose a function to differentiate, called 'u', and another function to integrate, called 'dv'. For this problem, we will choose: u = sec^12(x) dv = tan(x)dx
02

Differentiate 'u' and Integrate 'dv'

We need to differentiate 'u' and integrate 'dv' with respect to x: du/dx = d/dx(sec^12(x)) = 12sec^11(x)sec(x)tan(x), hence du = 12sec^11(x)sec(x)tan(x)dx Integrate 'dv': v = 鈭玹an(x)dx = ln|sec(x)| + C (constant C)
03

Apply Integration by Parts Formula

Apply the integration by parts formula: 鈭玼dv = uv - 鈭玽du 鈭玸ec^12(x)tan(x)dx = sec^12(x)ln|sec(x)| - 鈭�12sec^11(x)sec(x)tan(x)ln|sec(x)|dx
04

Simplify the Integral

Combine the factors with like terms and rewrite the integral: 鈭玸ec^12(x)tan(x)dx = sec^12(x)ln|sec(x)| - 12鈭玸ec^12(x)tan^2(x)ln|sec(x)|dx
05

Apply Trigonometric Identity

Use the trigonometric identity: tan^2(x) = sec^2(x) - 1 鈭玸ec^12(x)tan(x)dx = sec^12(x)ln|sec(x)| - 12鈭玸ec^12(x)(sec^2(x) - 1)ln|sec(x)|dx
06

Distribute and Break Into Two Integrals

Distribute the sec^12(x) inside the parenthesis and break the resulting expression into two integrals: 鈭玸ec^12(x)tan(x)dx = sec^12(x)ln|sec(x)| - 12(鈭玸ec^14(x)ln|sec(x)|dx - 鈭玸ec^12(x)ln|sec(x)|dx) At this point, the integral becomes quite complicated and may not have a simple closed-form solution. In practice, it's common to solve complicated integrals like this using numerical methods or approximations, depending on the purpose of the integral.

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are algebraic expressions involving trigonometric functions that are true for all values of the variables involved. They are crucial in simplifying trigonometric expressions and solving integrals, as seen in the exercise with \( \int \sec^{12} x \tan x \, dx \).
  • One of the key trigonometric identities used is \( \tan^2(x) = \sec^2(x) - 1 \). This simplifies expressions involving \( \tan(x) \) by rewriting them in terms of \( \sec(x) \), which can be more straightforward to integrate or manipulate.
  • Knowing these identities can help to transform complex trigonometric integrals into more manageable forms.
Differentiation
Differentiation is a core concept in calculus that deals with finding the derivative of functions. It is the process of determining the rate at which a function is changing at any given point.
  • For this particular exercise, differentiation is crucial when determining \( du \) in the integration by parts formula. We first choose \( u = \sec^{12}(x) \) and differentiate it.
  • The derivative of \( u \) is \( du = 12 \sec^{11}(x) \sec(x) \tan(x) \, dx \). This derivative is then used in the integration by parts formula to solve the integral.
Understanding how to differentiate trigonometric functions accurately is vital, as errors in differentiation can lead to incorrect solutions.
Integration Techniques
Integration techniques are various strategies used to evaluate integrals, especially those that are not straightforward. When it comes to integrals involving products of functions, integration by parts is a powerful technique.
  • Integration by parts is based on the product rule of differentiation and is useful for integrals of the form \( \int u \, dv = uv - \int v \, du \).
  • Choosing the right \( u \) and \( dv \) is key to simplifying the problem. In this case, choosing \( u = \sec^{12}(x) \) and \( dv = \tan(x) \, dx \) allows for a more manageable integration process, though this integral remains complex.
After applying integration by parts, it's common to encounter integrals that require additional techniques or identities, such as trigonometric substitutions or numerical methods, to evaluate.

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