   Chapter 7.3, Problem 10E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 2 / 3 4 − 9 x 2 d x

To determine

To evaluate: The given integral 02349x2dx.

Explanation

Integration involving terms of the form a2x2 can be simplified by using the trigonometric substitution x=asinθ.

Formula used:

The identity, cos2x=1sin2x

Given:

The integral, 02349x2dx

Calculation:

The integral can be written as:

02349x2dx=0239(49x2)dx=0233(2232x2)dx

Substitute for x as x=23sinθ. Take the derivative of the substitution term:

x=23sinθdx=23cosθdθ

Here, π2θπ2. The limits of integration will change as:

x023sinθ=0θ0andx2323sinθ=23sinθ=1θπ2

Substitute for x and dx in the given integral to get:

02349x2dx=0π23(22322232sin2θ)23cosθdθ=430π2(1sin2θ)cosθdθ

Use the identity <

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