IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

munch

How do i prove that _[a]

^[a+T] f(x) dx=_[0]

^[T] f(x) dx ?

aman23iit

hi friend its simple if a is the period of the function

then you know that

f(x-a)dx=

f(x)dx

now in _[a]^[a+T] f(x) dx put x-a =z and change the limit accordingly you will get _{0]^[T] f(z) dz

now put z=x you will get

_0]^[T] f(x) dx

bye plzz rate me

iitkgp_bipin

Using property of integrals :

$\begin{displaymath}\int_{c}^{c+T}f(x)dx=\int_{c}^{0}f(x)dx+\int_{0}^{T}f(x)dx+\int_{T}^{c+T}f(x)dx\mbox{.}\end{displaymath}$

In the last integral take
z = x - T
dz = dx
f(x) = f(z+T) = f(z)
as x varies from T to c+T , hence z would vary from 0 to c.

Hence

$\begin{eqnarray*}\int_{c}^{c+T}f(x)dx&=&\int_{c}^{0}f(x)dx+\int_{0}^{T}f(x)dx+\... ...c}f(x)dx+\int_{0}^{T}f(x)dx+\int_{0}^{c}f(z)dz=\int_{0}^{T}f(x)dx\end{eqnarray*}$

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