Question

To solve the integral ∫ᵃᶜᵒˢᶿ ᵈᶿ/√ₐ²₍₁₋ₛᵢₙ²θ₎, what substitution should be made?

a) Set x = asinθ
b) Set x = acosθ
c) Set x = atanθ
d) Set x = asecθ

Answer 1

The correct substitution for solving provided integral is x = asinθ (option a).

Step-by-step explanation:

The correct substitution for solving the integral ∫ᵃᶜᵒˢᶿ ᵈᶿ/√ₐ²₍₁₋ₛᵢₙ²θ₎ is to set x = asinθ (option a). This substitution helps in transforming the integral into a form that can be easily evaluated. Here are the steps:

1. Let x = asinθ.
2. Substitute x into the integral to get ∫asinθ cos(acos(asinθ)) d(asinθ)/√(a² - sin²θ).
3. Simplify the integral using trigonometric identities, such as cos(acos(asinθ)) = cosθ.
4. Apply the substitution in the limits of integration as well, if needed.
5. Evaluate the integral and simplify the result if required.