Question

Which of the following can be converted to the form ∫ (1/w) dw using a substitution?

(a) ∫ (cos(x) / (2 + sin(x))) dx
(b) ∫ (1 / √(2x + 1)) dx

Answer 1

Option (b) ∫ (1 / √(2x + 1)) dx can be converted to the form ∫ (1/w) dw using the substitution w = 2x + 1.

Step-by-step explanation:

To determine which of the given integrals can be converted to the form ∫ (1/w) dw using a substitution, we need to look for an expression in the integral that resembles the derivative of 1/w. In this case, option (b) ∫ (1 / √(2x + 1)) dx can be converted to the given form by using the substitution w = 2x + 1. Let's see the step-by-step process:

1. Let w = 2x + 1
2. Calculate dw/dx: dw/dx = 2
3. Solve for dx: dx = dw/2
4. Substitute the values of w and dx in the integral: ∫ (1 / √(2x + 1)) dx = ∫ (1 / √w) (dw/2)
5. Now, we can simplify the integral to: (1/2) ∫ (1/w) dw

Therefore, option (b) ∫ (1 / √(2x + 1)) dx can be converted to the form ∫ (1/w) dw using the substitution w = 2x + 1.