   Chapter 12.2, Problem 54E

Chapter
Section
Textbook Problem

# Evaluating a Definite Integral In Exercises 47-52, evaluate the definite integral. ∫ 0 π / 4 [ ( sec t tan t ) i + ( tan t ) j + ( 2 sin t cos t ) k ) ] d t

To determine

To calculate: The simplified value of the definite integral 0π/4[(secttant)i+(tant)j+(2sintcost)k]dt.

Explanation

Given:

The definite integral is 0π/4[(secttant)i+(tant)j+(2sintcost)k]dt.

Formula used:

If r(t)=f(t)i+g(t)j+h(t)k, where f, g, and h are continuous function on [a,b], then the definite integral of r is:

abr(t)dt=[abf(t)dt]i+[abg(t)dt]j+[abh(t)dt]k

Calculation:

Consider the following definite integral:

0π/4[(secttant)i+(tant)j+(2sintcost)k]dt

Evaluate the integration and substitute the limits.

Thus, obtain the following result:

0π/4[(secttant)i+(tant)j+(2sintcost)k]dt=[[0π/4(secttant)dt]i+[0π/4(tant)dt]j<

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