Question

2∫(0.01sen3πx)×(senπx)dx
interval: 0 - 1

Answer 1

the answer may not result in zero

Answer 2

$$\cos(2\pi x)=\cos(3\pi x-\pi x)=\cos(3\pi x)\cos(\pi x)+\sin(3\pi x)\sin(\pi x)\\\cos(4\pi x)=\cos(3\pi x+\pi x)=\cos(3\pi x)\cos(\pi x)-\sin(3\pi x)\sin(\pi x)$$

Answer 3

therefore \(\cos(2\pi x)-\cos(4\pi x)=2\sin(3\pi x)\sin(\pi x)\) allowing us to rewrite our integral:$$\begin{align*}0.01\int_0^1(\cos(2\pi x)-\cos(4\pi x))\ dx&=0.01\left(\frac1{2\pi}\sin(2\pi x)-\frac1{4\pi}\sin(4\pi x)\right)_0^1\\&=0.01\left(-\left(\frac1{2\pi}-\frac1{4\pi}\right)\right)\\&=-\frac1{400\pi}\end{align*}$$

Answer 4

oops actually I forgot \(\sin0=0\ne 1\) so actually the second line should read \(0.01\cdot0\) and our integral result is just \(0\) -- cool!

Answer 5

yes

Answer 6

we can use trigonometric identity

Answer 7

but my result is zero

Answer 8

this is because \(\sin(nx),\sin(mx)\) are orthogonal on \([0,\pi]\) for \(n,m\) both either odd or even

Answer 9

and yes the result is \(0\) I wrote the wrong answer down above the integral evaluates to \(0\)

Answer 10

but the result can not be zero

Answer 11

see for yourself: http://www.wolframalpha.com/input/?i=+2%E2%88%AB_0%5E1+%280.01sin%283%CF%80x%29%29%C3%97%28sin%28%CF%80x%29%29dx

Answer 12

but it is a wave equation, there must be an outcome

Answer 13

2∫(0.01sen3πx)×(senπnx)dx
interval: 0 - 1

Answer 14

have the letter N

Answer 15

okay if it has the letter \(n\) then the answer is \(0\) for all odd \(n=2k+1\) but the result is different for even \(n=2k\)

Answer 16

actually, it's *always* \(0\) for integers \(n,m\)!

http://www.wolframalpha.com/input/?i=int_0%5Epi+sin%28m+x%29+sin%28n+x%29+dx