Definite Integration
Integration
Grade Class 12
Question:
<p>Let <i>F</i>(<i>x</i>) be an indefinite integral of sin<sup>2</sup><i>x</i>.</p><p><b>Statement-1 :</b> The function <i>F</i>(<i>x</i>) satisfies <i>F</i>(<i>x</i> + π) = <i>F</i>(<i>x</i>) for all real <i>x</i>.</p><p><b>because</b></p><p><b>Statement-2 :</b> sin<sup>2</sup>(<i>x</i> + π) = sin<sup>2</sup><i>x</i> for all real <i>x</i>.</p>
(A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1.
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(C) Statement-1 is True, Statement-2 is False.
(D) Statement-1 is False, Statement-2 is True.
Step-by-Step Solution
Key Concept: An indefinite integral F(x) of a periodic function f(x) with period T is not necessarily periodic with the same period T. Specifically, F(x+T) = F(x) + C, where C is the integral of f(x) over one period.
<p>Statement-2 is true because sin<sup>2</sup>(<i>x</i> + π) = (-sin <i>x</i>)<sup>2</sup> = sin<sup>2</sup><i>x</i>. Let <i>F</i>(<i>x</i>) = ∫ sin<sup>2</sup><i>x</i> <i>dx</i> = ∫ (1 - cos 2<i>x</i>)/2 <i>dx</i> = <i>x</i>/2 - (sin 2<i>x</i>)/4 + <i>C</i>. Then <i>F</i>(<i>x</i> + π) = (<i>x</i> + π)/2 - (sin 2(<i>x</i> + π))/4 + <i>C</i> = <i>x</i>/2 + π/2 - (sin 2<i>x</i>)/4 + <i>C</i> = <i>F</i>(<i>x</i>) + π/2. Thus <i>F</i>(<i>x</i> + π) ≠ <i>F</i>(<i>x</i>). So Statement-1 is false.</p>
Correct Answer: 4