/*! 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 69 Use the reduction formulas in a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 69

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \tan ^{4} 3 y d y$$

Short Answer

Expert verified
Answer: The integral of \(\tan^{4}(3y) dy\) is \(\frac{1}{9}\tan^2(3y) - \frac{1}{3} + y + C\), where \(C\) is the integration constant.

Step by step solution

01

Identify the parameters

In this problem, we have \(n = 4\) (the power of tangent function), \(a = 3\) (the constant coefficient), and the variable \(x\) is replaced by \(y\).
02

Apply the reduction formula

Now, using the reduction formula, we get: $$\int \tan^{4}(3y) dy = \frac{1}{3} \cdot \frac{\tan^2(3y)}{3} - \int \tan^2(3y) dy$$
03

Simplify the integral

Simplify the first term on the right-hand side: $$\int \tan^{4}(3y) dy = \frac{1}{9}\tan^2(3y) - \int \tan^2(3y) dy$$
04

Apply the reduction formula again to the remaining integral

For the remaining integral of \(\tan^2(3y)\), we apply the reduction formula again, as we did before. This time, \(n = 2\), \(a = 3\), and the variable is still \(y\). So we get: $$\int \tan^2(3y) dy = \frac{1}{3} \cdot \frac{\tan^0(3y)}{1} - \int \tan^0(3y)dy$$ Now, \(\tan^0(3y) = 1\), so the integral becomes: $$\int \tan^2(3y) dy = \frac{1}{3} \cdot \frac{1}{1} - \int 1 dy$$
05

Simplify further and evaluate the remaining integral

Simplify the expression and evaluate the remaining integral: $$\int \tan^{2}(3y) dy = \frac{1}{3} - \left(\int 1 dy\right) = \frac{1}{3} - y$$
06

Substitute the result back into the original expression

Now, substitute the result from Step 5 back into the expression from Step 3: $$\int \tan^{4}(3y) dy = \frac{1}{9}\tan^2(3y) - \left(\frac{1}{3} - y\right)$$
07

Add the integration constant

Finally, add the integration constant, \(C\), to complete the solution: $$\int \tan^{4}(3y) dy = \frac{1}{9}\tan^2(3y) - \frac{1}{3} + y + C$$

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

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt[4]{x+2}+1} ; x+2=u^{4}$$

When is the volume finite? Let \(R\) be the region bounded by the graph of \(f(x)=x^{-p}\) and the \(x\) -axis, for \(x \geq 1.\) a. Let \(S\) be the solid generated when \(R\) is revolved about the \(x\) -axis. For what values of \(p\) is the volume of \(S\) finite? b. Let \(S\) be the solid generated when \(R\) is revolved about the \(y\) -axis. For what values of \(p\) is the volume of \(S\) finite?

Use numerical methods or a calculator to approximate the following integrals as closely as possible. $$\int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x=\frac{\pi}{2}$$

Bob and Bruce bake bagels (shaped like tori). They both make standard bagels that have an inner radius of 0.5 in and an outer radius of 2.5 in. Bob plans to increase the volume of his bagels by decreasing the inner radius by \(20 \%\) (leaving the outer radius unchanged). Bruce plans to increase the volume of his bagels by increasing the outer radius by \(20 \%\) (leaving the inner radius unchanged). Whose new bagels will have the greater volume? Does this result depend on the size of the original bagels? Explain.

Imagine that today you deposit \(\$ B\) in a savings account that earns interest at a rate of \(p \%\) per year compounded continuously. The goal is to draw an income of \(\$ I\) per year from the account forever. The amount of money that must be deposited is \(B=I \int_{0}^{\infty} e^{-n t} d t,\) where \(r=p / 100 .\) Suppose you find an account that earns \(12 \%\) interest annully and you wish to have an income from the account of \(\$ 5000\) per year. How much must you deposit today?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.