Hello everyone, I'm just having trouble on two questions for my Calculus work. I need to solve them using trig substitution to eliminate the root. Does anyone know where to start with this problem? Any help would be greatly appreciated!

The main idea is to exploit the trigonometric identity,

sin²(θ) + cos²(θ) = 1

2. For an integral containing 16 - 81x², you might substitute x = 4/9 sin(θ) (with differential dx = 4/9 cos(θ) dθ, but without an actual integral to work with this isn't really important). Then

16 - 81x² = 16 - 81 (4/9 sin(θ))²

… = 16 - 81 (16/81 sin²(θ))

… = 16 - 16 sin²(θ)

… = 16 (1 - sin²(θ))

… = 16 cos²(θ)

so that in the root expression, we would end up with

since for all real a, b, and c; ; and for all real x.

The goal is to replace x with some multiple of sin(θ) that makes the coefficients factor out like they did here, which then lets you reduce 1 - sin²(θ) to cos²(θ).

And don't be discouraged by the absolute values; in the context of a definite integral, there are things that can be done to remove them or otherwise simplify absolute value expressions.

3. Substitute z = 1/√8 sin(θ) (so that dz = 1/√8 cos(θ) dθ). Then

1 - 8z² = 1 - 8 (1/√8 sin(θ))²

… = 1 - 8 (1/8 sin²(θ))

… = 1 - sin²(θ)

… = cos²(θ)

so that

c. side angle side. u shares the same sides