Let T > 0 be a fixed real number. Suppose, f is a continuous function such that for all x ∈ R, f(x + T) = f(x). If I = int_0^T f(x) dx, then the value of int_3^3+3T f(2x) dx is

Let T > 0 be a fixed real number. Suppose, f is a… — JEE Main 2002 Mathematics PYQ

AIEEE 2002

Let

T > 0

be a fixed real number. Suppose,

f

is a continuous function such that for all

x \in R

f(x + T) = f(x)

. If

I = \int_0^T f(x)\, dx

, then the value of

\int_3^{3+3T} f(2x)\, dx

A.
$\frac{3}{2}I$
B.
$I$
C.
$3I$
D.
$6I$

Show answer and solution

Correct option: C

Solution

Substitute u = 2x, so du = 2dx. The integral becomes (1/2)∫₆^(6+6T) f(u) du.

Since f is periodic with period T, the integral of f over any interval of length nT equals n·I. The interval [6, 6+6T] has length 6T, so ∫₆^(6+6T) f(u) du = 6I.

Therefore the answer is (1/2)·6I = 3I.

[AIEEE 2002]

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