Definite Integration
Indefinite Integration
Grade Class 12
Question:
Let <i>F</i>(<i>x</i>) be an indefinite integral of sin<sup>2</sup><i>x</i>.<br><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>.<br><b>because</b><br><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>.
(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: The integral of a periodic function with period T is not necessarily periodic with the same period T; it may have a linear term.
<i>F</i>(<i>x</i>) = ∫ sin<sup>2</sup><i>x</i> d<i>x</i> = ∫ (1 - cos 2<i>x</i>)/2 d<i>x</i> = <i>x</i>/2 - (sin 2<i>x</i>)/4 + <i>C</i>. <br><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. <br>Thus, <i>F</i>(<i>x</i> + π) ≠ <i>F</i>(<i>x</i>). Statement-1 is False. <br>Statement-2 is True as sin<sup>2</sup>(<i>x</i> + π) = (-sin <i>x</i>)<sup>2</sup> = sin<sup>2</sup><i>x</i>.
Correct Answer: (B)