Question

Find the half range of cosine and sine expansion of:
f(x) = { 1, 0 10 years ago

Answer 1

The half range here is \(1\), so the cosine and sine expansions are given by
\[2\int_0^1f(x)\cos nx\,dx\quad\text{and}\quad2\int_0^1f(x)\sin nx\,dx\](respectively).

Compute these by splitting up the interval of integration; for example,
\[2\int_0^1f(x)\cos nx\,dx=2\left(\int_0^{1/2}\cos nx\,dx+\int_{1/2}^1 0\,dx\right)=2\int_0^{1/2}\cos nx\,dx\]
Can you take it from here?