Let f(x) be an indefinite integral of ^3 x . Statement 1: f(x) is a periodic function of period π . Statement 2: ^3 x is a periodic function.

Correct answer: D. Statement 1 is false, Statement 2 is true. Statement -2: ^3 x is a periodic function. it is a true statement. Statement-1 Given f(x)=∫ ^3 x d x aligned & =∫ (( 3 x)/(4)+(3 x)/(4))d x & =(1)/(4) ( 3 x)/(3)+(3)/(4) & =(1)/(12) 3 x+(3)/(4) aligned Now, period of (1)/(12) 3 x= (2 π)/(3) Period of (3)/(4) x=2 π Hence period of f(x)= L.C.M. (2 π, 2 π HCF of (1,3) =(2 π)/(1)=2 π Thus, f(x)is a periodic function of period 2 π. Hence, Statement -1 is false.

JEE Main 20127 MayOnline

Exam: JEE Main
Year: 2012
Date: 7 May
Mode: Online
Subject: Mathematics
Chapter: Indefinite Integration

JEE Main 2012 · 7 May · Online · Mathematics · Indefinite Integration

Let f(x) be an indefinite integral of ^3 x . Statement 1: f(x) is a...

Let

f(x)

be an indefinite integral of

\cos ^3 x

. Statement 1:

f(x)

is a periodic function of period

\pi

. Statement 2:

\cos ^3 x

is a periodic function.

Options

AStatement 1 is true, Statement 2 is false.
BBoth the Statements are true, but Statement 2 is not the correct explanation of Statement 1.
CBoth the Statements are true, and Statement 2 is correct explanation of Statement 1.
DStatement 1 is false, Statement 2 is true.

Correct answer

D. Statement 1 is false, Statement 2 is true.

Step-by-step solution

Statement

-2: \cos ^3 x

is a periodic function. it is a true statement. Statement-1 Given

f(x)=\int \cos ^3 x d x

\begin{aligned}& =\int \left(\frac{\cos 3 x}{4}+\frac{3 \cos x}{4}\right)d x \\ & =\frac{1}{4} \frac{\sin 3 x}{3}+\frac{3}{4} \sinx \\ & =\frac{1}{12} \sin3 x+\frac{3}{4} \sinx \end{aligned}

Now, period

\frac{1}{12} \sin3

\frac{2 \pi}{3}

Period of

\frac{3}{4} \sin x=2 \pi Hence period of f(x)=\frac{\text{ L.C.M. }(2 \pi, 2 \pi}{\text{ HCF of }(1,3)} =\frac{2 \pi}{1}=2 \pi Thus,

f(x)is a periodic function of

period 2 \pi. Hence, Statement

-1

is false.

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