You need to prove that question int_0^1 sin (pimx)sin(pinx)dx={(0m !ne n),(1/2m=n):}

Khaleesi Herbert 2021-01-19 Answered

You need to prove that question
01sin(πmx)sin(πnx)dx={(0mn1/2m=n)

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Expert Answer

Szeteib
Answered 2021-01-20 Author has 102 answers
We have to show
01sin(πmx)sin(πnx)dx={(0mn1/2m=n)
We will use the following trigonometric identities:
sinAsinB=12[cos(AB)cos(A+B)]
Thus we have
sin(πmx)sin(πnx)=12[cos(mn)πxcos(m+n)πx]
First consider the case mn. Then we have
01sin(πmx)sin(πnx)dx=0112[cos(mn)πxcos(m+n)πx]dx
=(sin[(mn)πx]2(mn)πsin[(m+n)πx]2(m+n)π)|01
=0,
because sin(kπ)=0, for all k in ZZ. Considering m=n than we have
01sin2(πmx)dx=0112[1cos2mπx]dx
=(x2sin[2mπx]4mπ)|01
=12,
And we get the finally answer is
01sin(πmx)sin(πnx)dx={(0mn1/2m=n)
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