**Integration of sin^-1(x)/x^2**

To integrate the function sin^-1(x)/x^2, we can use the method of integration by parts. This method involves breaking down the function into two parts and applying the integration formula. Let's go through the steps in detail:

**Step 1: Identify the Parts**
We can split the function sin^-1(x)/x^2 into two parts: sin^-1(x) and 1/x^2.

**Step 2: Choose u and dv**
In integration by parts, we choose one part as "u" and the other part as "dv". In this case, we can choose u = sin^-1(x) and dv = 1/x^2.

**Step 3: Find du and v**
To find du, we differentiate u with respect to x. In this case, du = 1/√(1-x^2) dx. To find v, we integrate dv with respect to x. In this case, v = -1/x.

**Step 4: Apply the Integration by Parts Formula**
The integration by parts formula states: ∫ u dv = uv - ∫ v du.

Applying this formula to our chosen u and dv, we have:

∫ sin^-1(x) * (1/x^2) dx = (-1/x) * sin^-1(x) - ∫ (-1/x) * (1/√(1-x^2)) dx

**Step 5: Simplify the Integral**
To simplify the integral, we multiply the terms inside the integral:

= (-1/x) * sin^-1(x) + ∫ 1/(x√(1-x^2)) dx

**Step 6: Rewrite the Integral**
To further simplify the integral, we can rewrite it using a trigonometric substitution. Let's substitute x = sinθ:

dx = cosθ dθ
√(1-x^2) = √(1-sin^2θ) = √(cos^2θ) = cosθ

The integral becomes:

∫ 1/(x√(1-x^2)) dx = ∫ 1/(sinθ * cosθ) * cosθ dθ
= ∫ cotθ dθ

**Step 7: Evaluate the Integral**
The integral of cotθ can be evaluated as ln|sinθ| + C, where C is the constant of integration.

Therefore, the final result is:

∫ sin^-1(x)/x^2 dx = (-1/x) * sin^-1(x) + ln|sinθ| + C

Remember to substitute back x = sinθ to obtain the answer in terms of x.

Overall, the integration of sin^-1(x)/x^2 involves applying the integration by parts formula, simplifying the integral, and using a trigonometric substitution to evaluate the integral.