Integrals

Calculus Level 4

Let f ( x ) = 1 2 a 0 + ∑ i = 1 n a i cos ⁡ ( i x ) + ∑ j = 1 n b j sin ⁡ ( j x ) f(x)=\dfrac{1}{2}a_0+\displaystyle\sum_{i=1}^n a_i \cos(ix)+\displaystyle\sum_{j=1}^nb_j \sin(jx) , then ∫ − π π f ( x ) cos ⁡ ( k x ) d x \displaystyle\int_{-\pi}^{\pi} f(x) \cos (kx) \, dx (where k ≤ n k \leq n ) is equal to which of the following?

a k a_k b k b_k π a k πa_k π b k πb_k

Anish Raj by anish raj , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

J Joseph
Feb 3, 2017

Fourier series

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This is a method to find the correct answer and not a rigorous, complete proof.Take k=0.Hence,the integrand becomes f(x).The integral of all the terms of the form b j ∗ s i n ( j x ) bj*sin(jx) will be zero as it is an odd function.Also,the integral of all the terms of the form a j ∗ c o s ( j x ) aj*cos(jx) will be zero as the indefinite integral of c o s ( j x ) cos(jx) is 0 between the limits -pi to pi.Hence the only term remaining in the integrand is (1/2)a0 the integral of which between the limits -pi to pi is pi times a0.When we put k=0 in the options,only one option corresponds to the result we obtained,hence we get the correct answer as pi times ak.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Why don't you use latex

anish raj - 5 years, 2 months ago

Check this

anish raj - 5 years, 2 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...