Definite Integration
Leibniz Rule / Differentiation Under Integral
Grade 12
Question:
<p>If \(F(x) = \displaystyle\int_x^{x^2}(t^2+1)\,dt\), find \(F'(x)\).</p>
<li>\(x^4+1-x^2-1\)</li>
<li>\(2x(x^4+1)-(x^2+1)\)</li>
<li>\(2x(x^4+1)+(x^2+1)\)</li>
<li>\(x^4-x^2\)</li>
Step-by-Step Solution
Key Concept: Leibniz Rule: d/dx \int_{g(x)}^{h(x)} f(t)dt = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)
<div class='solution'>
<p><strong>Leibniz Rule:</strong> $F'(x) = f(x^2)\cdot(x^2)' - f(x)\cdot(x)'$</p>
<p>where $f(t) = t^2+1$.</p>
<p>$$F'(x) = ((x^2)^2+1)\cdot 2x - (x^2+1)\cdot 1$$</p>
<p>$$= (x^4+1)\cdot 2x - (x^2+1)$$</p>
<p>$$= \boxed{2x(x^4+1)-(x^2+1)}$$</p>
</div>
Correct Answer: B