   Chapter 8.4, Problem 22E

Chapter
Section
Textbook Problem

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

To determine

To calculate: The value of indefinite integral x236x2dx.

Explanation

Given:

The provided integral is x236x2dx.

Formula used:

Power rule:

xndx=xn+1n+1+C

Trigonometric identity:

sin2θ+cos2θ=1sin2θ=12cosθ2sin2θ=2sinθcosθ

Integration of Cosine function is;

cosθdθ=sinθ+C

Calculation:

None of the basic integration rules apply here. Now to use the trigonometric substitution observe that 36x2 is in the form of a2u2.

Let x=6sinθ, then differentiate both sides with respect to x to get:

dx=6cosθdθ

Now by using the figure find the value of θ,

sinθ=x6=perpendicularhypotenusebase=36x2

The value of cosθ is;

cosθ=36x26θ=arcsinx4

Substituting 6sinθ for x in the integral x236x2dx as;

x236x2dx=36sin2θ36(6sinθ)26cosθdθ=36sin2θ3636sin2θ6cosθdθ=36sin2θ36(1sin2θ)6cosθdθ

Recall the formula of trigonometric property, sin2θ+cos2θ=1

Apply the trigonometric property and simplify:

136x2dx=366sin2θcosθ36(1sin2θ)dθ=366sin2θcosθ16

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