Evaluate the integrals in Exercises 91–102.
96. ∫dy/((arcsin y)(1-y²))

Verified step by step guidance

1. Recognize that the integral is of the form \(\int \frac{dy}{(\arcsin y)(1 - y^2)}\). Notice that the denominator contains \(1 - y^2\), which is related to the derivative of \(\arcsin y\).
2. Recall the derivative formula: \(\frac{d}{dy}(\arcsin y) = \frac{1}{\sqrt{1 - y^2}}\). This suggests a substitution involving \(\arcsin y\) might simplify the integral.
3. Let \(t = \arcsin y\). Then, differentiate both sides to find \(dy\) in terms of \(dt\): \(dy = \cos t \, dt\), since \(y = \sin t\) and \(dy/dt = \cos t\).
4. Rewrite the integral in terms of \(t\): substitute \(y = \sin t\) and \(dy = \cos t \, dt\). Also, express \(1 - y^2\) as \(1 - \sin^2 t = \cos^2 t\). The integral becomes \(\int \frac{\cos t \, dt}{t \cdot \cos^2 t} = \int \frac{dt}{t \cdot \cos t}\).
5. Now, the integral is \(\int \frac{dt}{t \cos t}\). From here, consider methods such as integration by parts or special functions to proceed, depending on the context or allowed techniques.

Key Concepts

Integration Techniques Involving Inverse Trigonometric Functions

Integrals containing inverse trigonometric functions like arcsin y often require substitution or recognition of derivative forms. Understanding how to manipulate these functions and their derivatives is essential to simplify and evaluate the integral.

Derivative of arcsin y

The derivative of arcsin y is 1/√(1 - y²). Recognizing this derivative helps in identifying substitution opportunities, especially when the integral contains expressions like 1/(1 - y²) or its powers.

Substitution Method in Integration

Substitution involves changing variables to simplify an integral. For this problem, setting u = arcsin y transforms the integral into a more manageable form, allowing the integral to be expressed in terms of u and evaluated more easily.