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MathematicsAssertion and reason+3 / -1Question 57 of 66

IIT-JEE 2007 · Paper 1, Mathematics question 57

Let F(x)F(x) be an indefinite integral of sin⁡2x\sin^2x.

STATEMENT-1: The function F(x)F(x) satisfies F(x+π)=F(x)F(x+\pi)=F(x) for all real xx.

because

STATEMENT-2: sin⁡2(x+π)=sin⁡2x\sin^2(x+\pi)=\sin^2x for all real xx.

Choose one option
Answer and worked solution

Official answer: D

Antiderivative. F(x)=x2−sin⁡2x4+CF(x)=\dfrac x2-\dfrac{\sin2x}4+C, so
F(x+π)=F(x)+π2≠F(x).F(x+\pi)=F(x)+\frac\pi2\ne F(x).
Statement 1 is False. A periodic integrand has a periodic antiderivative only when its integral over a period is zero.

Statement 2. sin⁡(x+π)=−sin⁡x\sin(x+\pi)=-\sin x, so sin⁡2(x+π)=sin⁡2x\sin^2(x+\pi)=\sin^2x: True.

(D)

Topics:Standard integrals
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