Definite Integration
Evaluation of definite integrals
Grade 12
Question:
<p>The value of \(\int_0^1 \frac{\sin^{-1} x}{\sqrt{1-x^2}} dx\) is</p>
<p>(a) 0</p>
<p>(b) \(\frac{3}{2}\)</p>
<p>(c) \(\frac{\pi^2}{22}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand has the form f(x)·f'(x), where f(x) = sin⁻¹x and f'(x) = 1/√(1-x²). Use substitution u = sin⁻¹x to transform this into a simple power integral.
<p><strong>Step 1:</strong> Recognize the derivative pattern. Notice that d/dx(sin⁻¹x) = 1/√(1-x²), so the integrand is sin⁻¹x · d/dx(sin⁻¹x).</p><p><strong>Step 2:</strong> Use substitution u = sin⁻¹x, so du = 1/√(1-x²) dx.</p><p><strong>Step 3:</strong> When x = 0: u = sin⁻¹(0) = 0. When x = 1: u = sin⁻¹(1) = π/2.</p><p><strong>Step 4:</strong> The integral becomes ∫₀^(π/2) u du = [u²/2]₀^(π/2) = (π/2)²/2 - 0 = π²/8.</p><p>∴ Answer: C (π²/8)</p>
Correct Answer: C