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

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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 \(\arcsin y\).

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.

Let \(t = \arcsin y\). Then, differentiate both sides to find \(dy\) in terms of \(dt\): \(dy = \cos t \, dt\), since \(y = \sin t\) and \(dy/dt = \cos t\).

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 \(\int \frac{\cos t \, dt}{t \cdot \cos^2 t} = \int \frac{dt}{t \cdot \cos t}\).

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.

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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.

Evaluate the integrals in Exercises 91–102.96. ∫dy/((arcsin y)(1-y²))