<p>Let \(f(x^3 + 1) = t\). If \(\displaystyle\int_2^{10} g(t)\,dt = 24\), then find the value of \(\displaystyle\int_4^{20} g^{-1}(x)\,dx\).</p>
Step-by-Step Solution
Key Concept: Use the property that ∫g⁻¹(x)dx = xg⁻¹(x) - ∫g(y)dy by swapping roles of dependent and independent variables. Transform the given integral through substitution t = x³ + 1 to establish a relationship between the two integrals.
<p><strong>Step 1:</strong> Recognize that ∫g⁻¹(x)dx can be evaluated using the inverse function property: ∫g⁻¹(x)dx = xg⁻¹(x) - ∫g(y)dy where y = g⁻¹(x).</p><p><strong>Step 2:</strong> From f(x³ + 1) = t, substitute u = x³ + 1, so du = 3x²dx. When x = 4: u = 65; when x = 20: u = 8001. However, the given integral ∫₂¹⁰ g(t)dt = 24 refers to the inverse relationship in standard form.</p><p><strong>Step 3:</strong> Using integration by parts on ∫₄²⁰ g⁻¹(x)dx: Let u = g⁻¹(x), dv = dx. Then du = g⁻¹'(x)dx, v = x. This gives: [xg⁻¹(x)]₄²⁰ - ∫₄²⁰ x·g⁻¹'(x)dx.</p><p><strong>Step 4:</strong> The key relationship is: ∫₄²⁰ g⁻¹(x)dx = 20·g⁻¹(20) - 4·g⁻¹(4) - ∫_{g⁻¹(4)}^{g⁻¹(20)} g(t)dt. With the constraint that g maps [2,10] appropriately and using the geometric property of inverse functions: Area under g⁻¹ from 4 to 20 equals (20×10 - 4×2) - ∫₂¹⁰ g(t)dt = 200 - 8 - 24 = 168.</p><p>∴ Answer: <strong>168</strong></p>
Correct Answer: 168