Question

Trick substitution: integral of (y)√1-y²)?

a. Trigonometric substitution
b. Partial tion decomposition
c. Integration by parts
d. Completing the square

Answer 1

For the integral of (y)√(1-y²), a trigonometric substitution is the best approach. The substitution y = sin(θ) simplifies the integral using the Pythagorean identity. After solving the resulting integral, one must back-substitute to return to the variable y.

Step-by-step explanation:

To evaluate the integral of (y)√(1-y²), a trigonometric substitution is advised. This technique is useful when integrating functions with a square root involving (1-u²), (a²-u²), or similar expressions. In this case, substituting y with sin(θ) can simplify the integral due to the identity sin²(θ) + cos²(θ) = 1, which corresponds to the integral's square root term.

The step-by-step method to solve the integral would be:

1. Make the substitution y = sin(θ), which implies dy = cos(θ)dθ.
2. Substitute into the integral to get ∫ cos(θ)√(1-sin²(θ)) × cos(θ)dθ.
3. Simplify using the Pythagorean identity 1-sin²(θ) = cos²(θ).
4. Evaluate the resulting integral and then back-substitute to return to y variables.

The content-loaded trick substitution: integral of (y)√(1-y²)? is an example of when trigonometric substitution is not just a potential solution approach but the most appropriate one given the form of the integral.