Let f is a continuous function for all x- and fx-fx-2T- T-0- If I-02Tfxdx- then -44-40T f x4dx is equal to

As, f(x)=f(x+2T); f(x) has period 2T Now, Let I_1=∫_4^4+40T f( x4)dx Put x=4y, dx= 4dy ⇒ I_1=4∫_1^1+10T f(x)dx=4×5∫_0^2T f(x)dx ∵ nbsp;∫_a^a+nT f(y)dy=n∫_0^T ...

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Standard XII

Mathematics

Properties of Set Operation

Let f is a co...

Question

Let $f$ is a continuous function for all $x \in R$ and $f (x) = f (x + 2 T), T > 0.$ If $I = 2 T \int 0 f (x) d x,$ then $4 + 40 T \int 4 f (\frac{x}{4}) d x$ is equal to

5 I

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10 I

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20 I

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40 I

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Solution

The correct option is C $20 I$
As, $f (x) = f (x + 2 T); f (x)$ has period $2 T$
Now, Let $I_{1} = 4 + 40 T \int 4 f (\frac{x}{4}) d x$
Put $x = 4 y, d x = 4 d y$
$\Rightarrow I_{1} = 4 1 + 10 T \int 1 f (x) d x = 4 \times 5 2 T \int 0 f (x) d x ∵ a + n T \int a f (y) d y = n T \int 0 f (y) d y$
$∴ I_{1} = 20 I$

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Let f is a continuous function for all x∈R and f(x)=f(x+2T),T>0. If I=2T∫0f(x)dx, then 4+40T∫4f(x4)dx is equal to

The correct option is C 20IAs, f(x)=f(x+2T);f(x) has period 2T Now, Let I1=4+40T∫4f(x4)dx Put x=4y,dx=4dy ⇒I1=41+10T∫1f(x)dx=4×52T∫0f(x)dx∵a+nT∫af(y)dy=nT∫0f(y)dy ∴I1=20I

Let $f$ is a continuous function for all $x \in R$ and $f (x) = f (x + 2 T), T > 0.$ If $I = 2 T \int 0 f (x) d x,$ then $4 + 40 T \int 4 f (\frac{x}{4}) d x$ is equal to