   Chapter 8.4, Problem 38E

Chapter
Section
Textbook Problem

Converting the Limits of Integration In Exercises 37-42, Evaluate the definite integral using(a) The given integration limits and (b) The limits obtained by Trigonometric Substitution. ∫ 0 3 2 1 ( 1 − t 2 ) 5 2 d t

To determine

To calculate: The value of the definite integral 0321(1t2)52dt.

Explanation

Given:

The given definite integral is 0321(1t2)52dt.

Calculation:

Because (1t2) is of the form (a2x2), use the trigonometric substitution t=sinθ.

The derivative of above relation is;

dt=cosθdθ

When t=0 the value of θ is;

0=sinθθ=0

When t=32 the value of θ is;

32=sinθθ=π3

So, the definite integration can be written as;

0321(1t2)52dt=0π31(1sin2θ)52cosθdθ=0π31cos5θcosθdθ=0π31cos4θdθ=0π3sec4;

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