   Chapter 8.3, Problem 69E

Chapter
Section
Textbook Problem

# Comparing 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 4 3 x tan 3 3 x   d x

(a)

To determine

To calculate: The value of indefinite integral sec43xtan33xdx in two different ways

Explanation

Given:

The indefinite integral:

sec43xtan33xdx

Formula used:

Formula for integrals:

xndx=1n+1xn+1

Calculation:

Consider the specified integral:

sec43xtan33xdx

Evaluate the integral in two different methods as follows:

Method 1:

In case the power is even and positive for the secant, then the secant-squared factor must be saved and the remaining factors be converted to tangents. Then expand and integrate as follows

Therefore

(sec43xtan33x)dx=sec23xtan33xsec23xdx(1+tan23x)tan33xsec23xdx

Let u=tan3xdu=3sec23xdx

Now,

(sec43xtan33x)dx=(1+u2)u3du3=13(u3+u5)du=13(u44+u66)+C=13(tan43x4+tan63x6)+C

Therefore, (sec43xtan33x)dx=112tan43x+118tan63x+C1

Method 2:

In case the power is odd and positive for the secant, then the secant-tangent factor must be saved and the remaining factors be converted to secanst

(b)

To determine

To graph: The integral obtained by each method to show that the results differ only by a constant by using graphing utility. (Ignore the integration constant)

(c)

To determine

To prove: The anti-derivative of graphs obtained by two methods differ only by a constant analytically.

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