Question

For each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. Enter the appropriate letter (A,B, or C) in each blank. DO NOT EVALUATE THE INTEGRALS.

A. x = 4 tan θ B. x = 4 sin θ
C. x = 4 sec θ
1. ∫▒√(x^2-16 dx) 2. ∫▒(x^2 dx)/√(16-x^2 )

Answer 1

For the integral ∫▒√(x^2-16 dx), the substitution x = 4 sec θ would be the most helpful.

For the integral ∫▒(x^2 dx)/√(16-x^2 ), the substitution x = 4 sin θ would be the most helpful.

Explanation:

For the integral ∫▒√(x^2-16 dx), we can use the substitution x = 4 sec θ. This is because when we substitute x = 4 sec θ, we get:

dx = 4 sec θ tan θ dθ, and

√(x^2-16) = √(16 sec^2 θ - 16) = 4 tan θ.

Substituting these into the original integral, we get:

∫▒√(x^2-16 dx) = ∫▒4tan^2 θ(4 sec θ tan θ dθ) = 16∫▒tan^3 θ sec θ dθ.

This integral can then be evaluated using techniques such as integration by parts or trigonometric identities.

For the integral ∫▒(x^2 dx)/√(16-x^2 ), we can use the substitution x = 4 sin θ. This is because when we substitute x = 4 sin θ, we get:

dx = 4 cos θ dθ, and

√(16-x^2) = √(16-16sin^2 θ) = 4cos θ.

Substituting these into the original integral, we get:

∫▒(x^2 dx)/√(16-x^2 ) = ∫▒(16sin^2 θ)(4cos θ dθ)/(4cos θ) = 16∫▒sin^2 θ dθ.

This integral can then be evaluated using the half-angle formula for sine or the power-reducing formula for sine.