Download the App!

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

Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_0^1 \tan^3 \left (\frac{\pi x}{6} \right)\ dx $ ; entry 69

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

$=\frac{1}{\pi}+\frac{6}{\pi} \ln \left(\frac{\sqrt{3}}{2}\right)$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

01:35

Use the Table of Integrals…

05:50

02:25

$1-4$ Use the indicated en…

02:32

Use the indicated entry in…

02:45

06:26

Use the Table of Integral…

03:56

05:30

02:17

04:21

05:25

04:04

Okay. This question wants us to evaluate this integral using a formula from the tape. So if we look at our integral table in the back of the book, it says tow. Look at number 69 which says, but the integral of tangent. Cute. Do you is equal to 1/2 tangent squared of you Plus Ellen. Of course I knew plus c, of course. So since we know this formula, what we want to do is turn original integral into that form. So let's make a U substitution let you equal. Hi, ex over six. So d'you is just pi over six d x or D X equals six over pi, do you? And now we're almost done. But since we have a definite integral, we have to transform our limits. So you of one equals pi times one divided by six So high over six and you have zero is pi times zero over six so still zero Okay, so now we're ready to transform arena girl. So now we have this new expression are integral now goes from zero to pi over six of tangent cubed you do you with our general factor off six over pie in front. So now we just evaluate this using our formula from the integral table. So we get 1/2 tangents squared of you Plus Ln of co sign you and we're evaluating this anti derivative from zero to pi over six. So if we plug those in, we get six over pi times 1/2 tangent squared pi over six plus Ellen of co sign of pirates six minus six over pi times 1/2 tangents squared zero plus Ellen of co sign of zero and tangent of 00 co Sign of zeros one and Ellen of 10 So this term vanishes. So then we're just left with six over pi times 1/2 tangent squared of pirate or six plus Ellen of co sign of pi over six and evaluating these we get six over pile front times. Well, tangents squared of pi over six is group three over three quantity squared plus Ln of route three over two. And what is this? Simplify too. Well, after multiplying through by the six over pie, we get one over pi plus six over pi Ln of route three over to as our final answer

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:11

'34_ You are observing a rocket launch from point _ 3000 feet from the …

02:53

'd. Find the value of the slope and explain what it means context: Be c…

02:07

'7) A person can order new car With choice of 12 possible colors, with …

03:04

'Color can indicate relationships between groups of characters: What in…

00:45

'Consider vatiant of the Stag Hunt game Suppose that instead of two the…

03:17

'The table presents four sets of data prepared by the statistician Fran…

07:16

'Assume that when adults with smartphones are randomly selected, 42% us…

06:36

'Exercise. A light on the ground is 30 feet away from building the ligh…

06:21

'Problem 2:90 Students went to the zoo, 3 had hamburger; milk and c…

01:07

'Consider the numbers 1, 2,3,4, &nc 5. How many three digits seqque…