Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \frac{\sqrt{9x^2 + 4}}{x^2}\ dx $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Foster Wisusik

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

04:20

Use the Table of Integrals…

03:15

02:06

02:42

05:43

02:48

03:06

03:50

Use the Table of Integral…

02:16

Okay, This question wants us to integrate this function using a table. So if we look in the back of the book, we see that the closest thing to this expression is this formula. So what we need to dio is somehow get this integral to look like this one. So we want a use squared as our function inside the square root. So we want you squared to be equal to nine x squared. So this means that you is equal to three x, so d'you is equal to three d X or D X equals d'you divided by three and some additional things that may come in handy. This means the X squared equals you squared, divided by nine. Okay, so now we can start substituting. So we have are integral of the square root of Well, nine x squared is what we call the you squared plus four, which is two squared, all divided by excess squared, which we said was you squared divided by nine times our chain roll factor of 1/3 d u. And this simplifies to nine divided by three outside the integral, so three times the integral of the square root of you squared plus four over you squared, do you? So now we can directly apply our formula and get three times negative square root you squared plus four divided by you Plus Ln of you plus square root you squared plus a sward But in this case, we already know what a square it iss So let's just write it in to save us a step plus c All right, now let's back Substitute everything so we know that you squared is nine x squared and you is three x so you can see a cancellation is gonna happen here plus the natural log of three x again So we can already see that the three and the 1/3 cancel over there, which gives us a final answer of negative square root nine X squared plus four about my ex, plus the natural log of three x plus the square root of nine x squared plus four plus c. And that's our final answer. Oh, but we need a factor of three here, remember? From right here

View More Answers From This Book

Find Another Textbook

00:32

Jeff uses 3 fifth- size strips to model 3/5. he wants to use tenth size stri…

00:24

How do you use an exponent to represent a number such as 16

01:00

Rebecca's buys some scarves that costs $5 each and 2 purses that cost $…

01:48

please help me with this math problem: Alice has 1/5 as many miniature cars …

01:09

PLEASE HELPSeth paints houses for a fee of $39 and an hourly rate of $13. Ma…

01:23

What is 20% of what number is 96?

00:13

Phong is rolling a number cube and flipping a coin .what is the probability …

02:18

Name a pair of fractions that use the least common denominator and are equiv…

00:57

Peter put $8,000 into a savings account that pays 6% interest, compounded co…

01:26

The new number, 200, is 400 less than the original number. What is the appro…