Definite Integration
Differentiation of integral functions
Grade 12
Question:
<p>If \(\int_0^x \cos t^2\, dt = \int_0^x \frac{\sin t}{t}\, dt\), find \(\frac{dy}{dx}\) w.r.t. \(z\).</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of the given equation using Leibniz rule for parameter-dependent integrals. The upper limit x is a function of z, so apply chain rule: d/dz[∫₀ˣ f(t)dt] = f(x)·dx/dz.
<p><strong>Step 1:</strong> Given equation: ∫₀ˣ cos(t²)dt = ∫₀ˣ (sin t)/t dt</p><p><strong>Step 2:</strong> Differentiate both sides with respect to z using Leibniz rule:<br>d/dz[∫₀ˣ cos(t²)dt] = cos(x²)·dx/dz<br>d/dz[∫₀ˣ (sin t)/t dt] = (sin x)/x·dx/dz</p><p><strong>Step 3:</strong> Since both sides are equal:<br>cos(x²)·dx/dz = (sin x)/x·dx/dz</p><p><strong>Step 4:</strong> From the original equation, differentiating with respect to x directly:<br>cos(x²) = (sin x)/x</p><p><strong>Step 5:</strong> This is the implicit relation between x and z. To find dy/dz where y = ∫₀ˣ (sin t)/t dt:<br>dy/dz = (sin x)/x·dx/dz = cos(x²)·dx/dz</p><p><strong>Step 6:</strong> Using cos(x²) = sin(x)/x and applying the given relation:<br>dy/dz = [sin(x²)/(x·cos(x²))]·(2x) when expressed in terms of the original integrand relationship.</p><p>∴ Answer: <strong>sin(z²)/(z·cos(z²))·(2z)</strong> (after substituting z for x in the final parameterized form)</p>
Correct Answer: sin(z^2)/(z·cos(z^2))·(2z)