   Chapter 8.4, Problem 37E

Chapter
Section
Textbook Problem

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral usingThe given integration limits and (b) the limits obtained by trigonometric substitution. ∫ 2 3 t ( 1 − t 2 ) 3 / 2 d t

To determine

To calculate: The value of the given definite integral 032t2(1t2)32dt.

Explanation

Given:

The given definite integral is 032t2(1t2)32dt.

Calculation:

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

The derivative of above is;

dt=cosθdθ

When t=0 the value of θ is;

0=sinθθ=0

When t=32 the value of θ is;

32=sinθπ3θ=π3

So, the definite integration can be written as;

032t2(1t2)32dt=0π3sin2

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