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MathCalculus

Integral of Trig Inverse Functions: Evaluate an Arcsine Integral

Recognize the arcsine derivative in the denominator, substitute u = arcsin t, and obtain the antiderivative 1/2(arcsin t)^2 + C with its real-domain boundary.

Question

In the following exercise, compute the antiderivative using an appropriate substitution:

∫sin⁡−1t1−t2 dt.\int \frac{\sin^{-1} t}{\sqrt{1-t^2}}\,dt.

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Evidence boundary

The notation follows OpenStax Exercise 411: sin⁻¹t is the principal inverse sine, not 1/sin t. For a real-valued antiderivative, the integrand is defined on −1<t<1; the endpoints make the derivative factor singular. The arbitrary constant C is required.