Question

Find an, in the function f(x) = x - x3
​

Answer 1

For the function f(x) = x - x³ the value of aₙ = 0

Correct question : Find aₙ for the function f(x) = x - x³

Given :

The function f(x) = x - x³

To find :

The value of aₙ

Solution :

Step 1 of 3 :

Check the function is even or odd

Here the given function is

$\displaystyle \sf{ f(x) = x - {x}^{3} }$

Now ,

$\displaystyle \sf{ f( - x) }$

$\displaystyle \sf{ = - x - {( - x)}^{3} }$

$\displaystyle \sf{ = - x + {x}^{3} }$

$\displaystyle \sf{ = - (x - {x}^{3})}$

$\displaystyle \sf{ = - f(x) }$

So the function f(x) = x - x³ is an odd function

Step 2 of 3 :

Define Fourier Series of a function

For a function f(x) in the interval ( - c, c)

The Fourier series is given by

$\displaystyle \sf{ f(x) = \frac{a_0}{2} + \sum\limits_{n=1}^{ \infty } a_n \: cos \frac{n\pi x}{c} + \sum\limits_{n=1}^{ \infty } \: b_n \: sin \frac{n\pi x}{c} }$

Where ,

$\displaystyle \sf{ a_0 = \frac{1}{c} \int\limits_{- c}^{c} f(x) \, dx }$

$\displaystyle \sf{ a_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: cos\frac{n\pi x}{c} \, dx }$

$\displaystyle \sf{ b_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx }$

Step 3 of 3 :

Calculate the value of aₙ

The given function f(x) = x - x³ is an odd function

Thus we get

$\displaystyle \sf{ a_0 = \frac{1}{c} \int\limits_{- c}^{c} f(x) \, dx } = 0$

$\displaystyle \sf{ \because \: cos \: \frac{n \pi x}{c} \: \: is \: an \: even \: function }$

$\displaystyle \sf{ \therefore \: f(x)cos \: \frac{n \pi x}{c} \: \: is \: an \: odd \: function }$

$\displaystyle \sf{ \therefore \: a_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: cos\frac{n\pi x}{c} \, dx } = 0$

Again ,

$\displaystyle \sf{ \because \: sin \: \frac{n \pi x}{c} \: \: is \: an \: odd \: function }$

$\displaystyle \sf{ \therefore \: f(x)sin \: \frac{n \pi x}{c} \: \: is \: an \: even \: function }$

$\displaystyle \sf{ \therefore \: b_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx = \frac{2}{c} \int\limits_{0}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx }$

Hence for the function f(x) = x - x³ the value of aₙ = 0

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