Question

Evaluate the indefinite integral. 1/(x²(sqrt(64 - x²))).

a) (1/8)arcsin(x/8)
b) (1/2)arcsin(x/8)
c) (1/4)arcsin(x/8)
d) (1/16)arcsin(x/8)

Answer 1

To evaluate the indefinite integral, we can use a trigonometric substitution. After simplifying and applying the partial fraction decomposition, the integral is expressed as the sum of two terms. Each term can be integrated using arcsin. Therefore, the correct option is (b) (1/2)arcsin(x/8).

Step-by-step explanation:

To evaluate the indefinite integral ∫ (1/(x²(sqrt(64 - x²)))) dx, we can start by using a trigonometric substitution. Let x = 8sin(θ) so that dx = 8cos(θ) dθ. Substituting these values, the integral becomes:

∫(1/((64sin²(θ))(8cos(θ)))) (8cos(θ) dθ

Simplifying further, we get:

(1/8) ∫(1/sin²(θ)) dθ

Using the identity sin²(θ) = 1 - cos²(θ), the integral becomes:

(1/8) ∫(1/(1 - cos²(θ))) dθ

Applying the partial fraction decomposition method, we can rewrite the integral as:

(1/16) ∫((1/(1 + cos(θ))) + (1/(1 - cos(θ)))) dθ

Now, we can integrate each term separately. The integral of (1/(1 + cos(θ))) can be expressed as (1/2) arcsin((sin(θ/2))/(cos(θ/2))) and the integral of (1/(1 - cos(θ))) can be expressed as (1/2) arcsin((sin(θ/2))/(cos(θ/2))). Therefore, the correct option for the given question is (b) (1/2) arcsin(x/8).