/*! 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 89 Evaluate the following definite ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 89

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 3} \sec x \tan x d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral \(\int_{0}^{\pi / 3} \sec x \tan x d x\). Answer: The definite integral is equal to 1.

Step by step solution

01

Find the antiderivative of \(\sec x \tan x\).

Recall that the derivative of the secant function (\(\sec x\)) is \(\sec x \tan x\). Therefore, the antiderivative of \(\sec x \tan x\) is \(\sec x\). Let's call this function F(x), so \(F(x) = \sec x\).
02

Evaluate F(x) at the endpoints.

Now we need to evaluate F(x) at the given endpoints, \(\pi / 3\) and \(0\). To do this, find the value of \(\sec (\pi / 3)\) and \(\sec (0)\): $$F(\pi / 3) = \sec (\pi / 3) = 2$$ $$F(0) = \sec (0) = 1$$
03

Calculate the difference between the evaluated antiderivative function.

Lastly, we find the difference, F(b) - F(a): $$\int_{0}^{\pi / 3} \sec x \tan x d x = F(\pi / 3) - F(0) = 2 - 1 = 1$$ So, the value of the definite integral is 1.

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

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=2-|x| ;[-2,4]$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s$$

The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval \([0, t],\) where \(0 \leq t \leq 3\). $$v(t)=\left\\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq 1.5 \\ 50 & \text { if } 1.5 < t \leq 3 \end{array}\right.$$

\(\text { Simplify the following expressions.}\) $$\frac{d}{d x} \int_{e^{x}}^{e^{2 x}} \ln t^{2} d t$$

Fill in the blanks with right, left, or midpoint; an interval; and a value of \(n\). In some cases, more than one answer may work. \(\sum_{k=1}^{8} f\left(1.5+\frac{k}{2}\right) \cdot \frac{1}{2} \mathrm{is} \mathrm{a}\) is a ________ Riemann sum for \(f\) on the interval \({____,_____]\) with \(n=\) ________.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.