/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Find the integral involving seca... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 21

Find the integral involving secant and tangent. $$ \int \sec ^{3} \pi x d x $$

Short Answer

Expert verified
The integral \( \int \sec^3 \pi x dx \) equals \( \frac{\pi}{2} \sec \pi x \tan \pi x + \frac{1}{2} \ln| \sec \pi x + \tan \pi x | + C \).

Step by step solution

01

Set up the integral by parts

Following the formula for integration by parts: \[ \int u dv = uv - \int v du \], set \( u = \sec x \) and \( dv = \sec^2 x dx \). Then, differentiate and integrate to find \( du = \sec x \tan x dx \) and \( v = \tan x \]. The integral now becomes: \[ \int \sec^3 x dx = \sec x \tan x - \int \tan^2 x \sec x dx\].
02

Use the Pythagorean identity

Remembering that \( \tan^2 x = \sec^2 x - 1 \), substitute it in the integral to get: \[ \int \sec^3 x dx = \sec x \tan x - \int (\sec^2 x \sec x - \sec x) dx\]. This can be further simplified to: \[ \sec x \tan x - \int \sec^3 x dx + \int \sec x dx \]
03

Solve for the integral

Rearranging the equation to isolate the unsolved integral, get: \[ \int \sec^3 x dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \int \sec x dx \]. The integral of \( \sec x \) is \( \ln | \sec x + \tan x | \) and therefore, the integral of \( \sec^3 x dx \) is: \[ \frac{1}{2} \sec x \tan x + \frac{1}{2} \ln | \sec x + \tan x | + C \].
04

Substitute \( \pi x \) back into the function

Lastly, as the original function is \( \sec^3 \pi x \), substitute \( \pi x \) back into the function and multiply with \( \pi \) to account for chain rule. The final answer is: \[ \frac{\pi}{2} \sec \pi x \tan \pi x + \frac{1}{2} \ln | \sec \pi x + \tan \pi x | + C \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integration by Parts
Integration by parts is a crucial technique in integral calculus used to solve integrals where other methods fail. It's particularly useful when dealing with products of functions. The formula is straightforward:
  • \( \int u \, dv = uv - \int v \, du \)
Here's how it works: you choose one part of your integrand as \( u \) and the other as \( dv \). Then, you differentiate \( u \) to get \( du \) and integrate \( dv \) to get \( v \). Substituting back into the equation helps solve the integral. For example, in the exercise, we set \( u = \sec x \) and \( dv = \sec^2 x \, dx \). This led us through a series of steps to eventually find our integral solution.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are vital in simplifying complex expressions, such as those encountered in calculus.The Pythagorean identity, \( \tan^2 x = \sec^2 x - 1 \), is specifically helpful here. It allows us to simplify expressions involving tangent and secant functions. Using this identity, we're able to switch terms to alternate but equivalent forms, sometimes making integration more straightforward. In the step-by-step solution, this identity was key to transforming the integral into a form where other methods could be applied.
Definite Integrals
Definite integrals are used to calculate the area under the curve of a function over a specified interval. Their notation typically looks like this:
  • \( \int_{a}^{b} f(x) \, dx \)
Here, \( a \) and \( b \) are the lower and upper bounds, respectively. However, the exercise we are tackling does not give specific bounds, implying it's an indefinite integral focusing on finding a general solution.Definite integrals would require additional steps beyond the symbolic antiderivative, involving the evaluation of the resulting antiderivative at the specified bounds and then subtracting these values. But remember, every indefinite integral can provide the building blocks you need for evaluating definite integrals effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that \(I_{n}=\left(\frac{n-1}{n+2}\right) I_{n-1},\) where \(I_{n}=\int_{0}^{\infty} \frac{x^{2 n-1}}{\left(x^{2}+1\right)^{n+3}} d x, \quad n \geq 1 .\) Then evaluate each integral. (a) \(\int_{0}^{\infty} \frac{x}{\left(x^{2}+1\right)^{4}} d x\) (b) \(\int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{5}} d x\) (c) \(\int_{0}^{\infty} \frac{x^{5}}{\left(x^{2}+1\right)^{6}} d x\)

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int \tan ^{4} \frac{x}{2} \sec ^{4} \frac{x}{2} d x $$

Think About It In Exercises 55-58, L'Hopital's Rule is used incorrectly. Describe the error. \(\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}}=\lim _{x \rightarrow 0} \frac{2 e^{2 x}}{e^{x}}=\lim _{x \rightarrow 0} 2 e^{x}=2\)

In Exercises 59 and \(60,\) (a) explain why L'H么pital's Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.