/*! 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 15 Use integration tables to find t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 15

Use integration tables to find the integral. $$ \int \frac{\cos \theta}{3+2 \sin \theta+\sin ^{2} \theta} d \theta $$

Short Answer

Expert verified
\[ \frac{1}{\sqrt{5}}\arctan\left(\frac{2\sin\theta+1}{\sqrt{5}}\right) + C \]

Step by step solution

01

Identify suitable substitution

First, it's necessary to note that if \(u = \sin \theta\), the expression inside the integral becomes simpler. So, let's let \(u = \sin \theta\), then \(du = \cos \theta d \theta\). This will allow us to rewrite the integral.
02

Apply substitution

Apply the substitution \(u = \sin \theta\), then \(du = \cos \theta d \theta\). So, the original integral can be rewritten as: \[ \int \frac{1}{3+2u+u^{2}} d u \]
03

Use the Integral Tables

The integral is now in a form that can be found using standard integral tables. The integral of \(\frac{1}{3+2u+u^{2}}\) with respect to \(u\) is \(\frac{1}{\sqrt{5}}\arctan\left(\frac{2u+1}{\sqrt{5}}\right) + C\). Here, \(\arctan(x)\) is the inverse tangent function and \(C\) is the constant of integration.
04

Substitute u back in

Now it's time to substitute \(u = \sin \theta\) back into the solution. This yields the final answer of: \[ \frac{1}{\sqrt{5}}\arctan\left(\frac{2\sin\theta+1}{\sqrt{5}}\right) + C \]

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!

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

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cosh a t $$

Use integration by parts to verify the reduction formula. $$ \int \sin ^{n} x d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x d x $$

Find the area of the region bounded by the graphs of the equations. $$ y=\cos ^{2} x, \quad y=\sin x \cos x, \quad x=-\pi / 2, \quad x=\pi / 4 $$

Describe the different types of improper integrals

Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.