Definite Integration
Differentiation Under Integral Sign
Grade 12
Question:
<p>If ∫_{sin <i>x</i>}^1 <i>t</i>² <i>f</i>(<i>t</i>) <i>d</i><i>t</i> = 1 − sin <i>x</i> for all <i>x</i> ∈ (0, π/2), then <i>f</i>(1/3) is:</p>
<p>(a) √3</p>
<p>(b) √3/2</p>
<p>(c) 1/3</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use Leibniz rule for differentiation under the integral sign to find the unknown function.
<p>Differentiate both sides with respect to <i>x</i> using Leibniz rule:<br/>−cos <i>x</i> · sin² <i>x</i> · <i>f</i>(sin <i>x</i>) = −cos <i>x</i><br/>This gives sin² <i>x</i> · <i>f</i>(sin <i>x</i>) = 1 for cos <i>x</i> ≠ 0.<br/>Let <i>u</i> = sin <i>x</i>; then <i>u</i>² <i>f</i>(<i>u</i>) = 1, so <i>f</i>(<i>u</i>) = 1/<i>u</i>².<br/>Therefore, <i>f</i>(1/3) = 1/(1/9) = 9. (Or verify by substitution method.)</p>
Correct Answer: A