Blog

Parin Sir > Mathematical Proofs > Definite Integration > Pattern – 20 : Definite integration > Pattern – 20 : Definite integration

Pattern – 20 : Definite integration

No Comments

⋆ Pattern – 20 :

If f(x) is an odd periodic defined on interval \left[-\frac{T}{2},\ \ \frac{T}{2}\right] , where T is a principal period of f(x), then  g (x) = \int_{a}^{x}\left(t\right) dt  is periodic function with principal period T.

i.e.  g (x + T) = g (x)

Proof :

Let, g(x) = \int_{a}^{x}f\left(t\right)dt

Now,

9\left(x+T\right)=\ \int_{a}^{x+T}f\left(t\right)\ dt

=\ \int_{a}^{x}f\left(t\right)dt+\ \int_{x}^{x+T}f\left(t\right)dt

=9\left(x\right)+\ \int_{0}^{T}f\left(t\right)dt

= 9(x) + 0

= 9(x)

 

Click here to see next pattern

Share
math of maths mathematics ganit proof definition formula any Definite integration pattern sabiti

Leave a Reply