   Chapter 8.4, Problem 21E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral In Exercises 21–36, find the indefinite integral. ∫ 1 16 − x 2 d x

To determine

To calculate: The value of indefinite integral 116x2dx.

Explanation

Given:

The provided integral is 116x2dx.

Formula Used:

Trigonometric identity:

sin2θ+cos2θ=1

Calculation:

None of the basic integration rules apply here. Now to use the trigonometric substitution observe that 16x2 is in the form of a2u2.

Let x=4sinθ, now differentiating on both sides with respect to x to get dx=4cosθdθ

Now, with the help of given figure find the value of θ,

sinθ=x4=perpendicularhypotenusebase=16x2

Therefore, θ=arcsinx4, substitute 4sinθ for x as;

116x2dx=4cosθ16(4sinθ)2dθ=4cosθ1616sin2θd

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