/*! 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 5 Evaluate the integrals in Exerci... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 5

Evaluate the integrals in Exercises \(1-22\) $$ \int \sin ^{3} x d x $$

Short Answer

Expert verified
\(-\cos x - \frac{\cos^3 x}{3} + C\)

Step by step solution

01

Identify the Problem Type

We need to evaluate \( \int \sin^3 x \ dx \). This is an example of an integral involving an odd power of sine, which suggests using a trigonometric identity for simplification.
02

Use Trigonometric Identity

We can use the identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite the integrand. Hence, \( \sin^3 x = \sin x \cdot \sin^2 x = \sin x \cdot (1 - \cos^2 x) \).
03

Substitute the Identity

We rewrite the integral as: \[ \int \sin^3 x \ dx = \int \sin x (1 - \cos^2 x) \ dx = \int (\sin x - \sin x \cos^2 x) \ dx. \]
04

Break Down into Simpler Integrals

This leads to two separate integrals: 1. \( \int \sin x \ dx \) 2. \( \int \sin x \cos^2 x \ dx \)
05

Solve the First Integral

\( \int \sin x \ dx \) is a basic integral resulting in \(-\cos x + C_1\).
06

Use Substitution for Second Integral

For \( \int \sin x \cos^2 x \ dx \), use substitution. Let \( u = \cos x \), then \( du = -\sin x \ dx \). The integral becomes: \[ -\int u^2 \ du \].
07

Integrate with Respect to \( u \)

Calculate the integral: \[ -\int u^2 \ du = -\left( \frac{u^3}{3} \right) + C_2 = -\frac{\cos^3 x}{3} + C_2. \]
08

Combine Results and Write Final Answer

Combine the results from previous steps: \[ -\cos x - \frac{\cos^3 x}{3} + C. \] Combining constants, the final answer is \(-\cos x - \frac{\cos^3 x}{3} + C\).

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.

Trigonometric Substitution
When encountering integrals involving trigonometric functions like \(\int \sin^3 x \ dx\), one effective technique is **trigonometric substitution**. This allows us to simplify the integral using known trigonometric identities. - **Identify the Function Type:** In examples with odd powers of sine or cosine, a strategic simplification can make the problem manageable.- **Utilize Trigonometric Identities:** For this exercise, we recognize that \( \sin^2 x = 1 - \cos^2 x \). This allows us to express \(\sin^3 x\) as \(\sin x \cdot (1 - \cos^2 x)\).By substituting, we break down the integral into simpler parts that can be individually solved, significantly reducing complexity.
Integral Calculus
**Integral Calculus** is the branch of calculus that deals with finding the accumulation of quantities, like areas under curves. In this context, we aim to evaluate the integral \(\int \sin^3 x \ dx\).- **Breaking Down the Integral:** After substitution, we split the original integral into two separate, simpler integrals: * \(\int \sin x \ dx\) * \(\int \sin x \cos^2 x \ dx\)- **Basic Integral Solutions:** The first integral, \(\int \sin x \ dx\), is a standard problem with a known solution of \(-\cos x + C_1\).- **More Complex Integrals:** The second requires further manipulation involving substitution, like setting \(u = \cos x\) to solve \(-\int u^2 \, du\).These steps transform a seemingly challenging problem into a series of manageable ones, each with a straightforward solution.
Trigonometric Identities
**Trigonometric Identities** play a crucial role in simplifying calculus problems involving trigonometric functions.- **Simplification of Integrals:** Using identities such as \(\sin^2 x = 1 - \cos^2 x\), enable the transformation of complex expressions into simpler forms.- **Strategy in Integration:** For the given problem, re-expressing \(\sin^3 x\) in terms of sine and cosine allows integrals to be broken down into solvable parts without initially overwhelming substitution or differentiation.Understanding these identities and their applications not only simplifies integrals but also builds a deeper intuition for calculus, making it an indispensable tool for solving complex problems.

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

Sine-integral function The integral $$\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t,$$ called the sine-integral function, has important applications in optics. \(\begin{equation} \begin{array}{l}{\text { a. Plot the integrand }(\sin t) / t \text { for } t>0 . \text { Is the sine-integral }} \\ \quad {\text { function everywhere increasing or decreasing? Do you think }} \\ \quad {\text { Si }(x)=0 \text { for } x>0 ? \text { Check your answers by graphing the }} \\ \quad {\text { function Si }(x) \text { for } 0 \leq x \leq 25 .} \\ {\text { b. Explore the convergence of }}\end{array} \end{equation}\) $$\int_{0}^{\infty} \frac{\sin t}{t} d t.$$ If it converges, what is its value?

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{2} \frac{d x}{1-x}$$

Suppose \(f\) is a probability density function for the random variable \(X\) with mean \(\mu .\) Show that its variance satisfies $$ \operatorname{Var}(X)=\int_{-\infty}^{\infty} X^{2} f(X) d X-\mu^{2} $$

Evaluate the integrals. \begin{equation} \quad \text { a. }\int_{0}^{1} \frac{d t}{\sqrt{t}(1+t)} \quad \text { b. } \int_{0}^{\infty} \frac{d t}{\sqrt{t}(1+t)}\end{equation}

In Exercises \(11-22,\) estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(11-18\) are the integrals from Exercises \(1-8 .\) . $$ \int_{-2}^{0}\left(x^{2}-1\right) d x $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.