   Chapter 8.3, Problem 68E

Chapter
Section
Textbook Problem

EXPLORING CONCEPTSComparing Methods In Exercises 67 and 68, (a) find the indefinite integral in two different ways (b) use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant, and (c) verify analytically that the results differ only by a constant. ∫ sec 2   x   tan   x   d x

(a)

To determine

To calculate:

The indefinite integral in two different ways

Explanation

Given:

sec2xtanxdx

Formula used:

(sec2xtanx)dx=(u)du

Calculation:

Consider the integral sec2xtanxdx

Need to evaluate the integral in two different methods.

Method 1:

When the power of the secant is even and positive, save a secant-squared factor and convert the remaining factors to tangents. Then expand and integrate as follows

Here there is only one secant-squared factor, so

(sec2xtanx)dx=tanx(sec2xdx)

Let u=tanxdu=sec2xdx

Now,

(sec2xtanx)dx

(b)

To determine

To graph:

The anti derivative (without the constant of integration) obtained by each method to show that the results differ only by a constant by using graphing utility

(c)

To determine

Analytically that the anti-derivative of graphs obtained by two methods differ only by a constant

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