7–84. Evaluate the following integrals.
79. ∫ (arcsinx)/x² dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves the function arcsin(x) divided by x². This suggests that integration by parts might be a useful technique. Recall the formula for integration by parts: ∫u dv = uv - ∫v du.
2. Step 2: Choose u = arcsin(x) and dv = 1/x² dx. This choice is made because the derivative of arcsin(x) simplifies nicely, and 1/x² dx is straightforward to integrate.
3. Step 3: Compute du by differentiating u = arcsin(x). The derivative of arcsin(x) is 1/√(1-x²), so du = (1/√(1-x²)) dx.
4. Step 4: Compute v by integrating dv = 1/x² dx. The integral of 1/x² is -1/x, so v = -1/x.
5. Step 5: Substitute u, v, du, and dv into the integration by parts formula: ∫(arcsin(x)/x²) dx = uv - ∫v du. This becomes (-arcsin(x)/x) - ∫(-1/x)(1/√(1-x²)) dx. Simplify and proceed to evaluate the remaining integral.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. In this context, evaluating the integral of a function like (arcsin x)/x² requires knowledge of techniques such as substitution or integration by parts.

Arcsine Function

The arcsine function, denoted as arcsin(x), is the inverse of the sine function, providing the angle whose sine is x. It is defined for values of x in the range [-1, 1] and outputs angles in radians between -π/2 and π/2. Understanding the properties and behavior of the arcsine function is crucial when integrating expressions that involve it, as it can affect the choice of integration techniques.

Improper Integrals

An improper integral is an integral that has either infinite limits of integration or an integrand that approaches infinity within the interval of integration. In the case of ∫ (arcsin x)/x² dx, it is important to determine if the integral converges or diverges, especially near points where x approaches 0 or infinity. Evaluating improper integrals often involves taking limits to ensure the integral is well-defined.