prove that the integral ∫ sin mx sin nx dx ={o if m≠n π if m=n where m and n are positive integers
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Step-by-step explanation:
−π
sinmxsinnxdx is 0 m≠n and π if m=n using integration by parts
calculus integration definite-integrals
Show that
∫
π
−π
sinmxsinnxdx={ 0 if m≠n, π if m=n.
by using integration by parts.
I've done the following, but I'm not sure if I went the wrong direction, if I messed up some calculation, or if I'm almost there and just can't see what to do next...
∫
π
−π
sinmxsinnxdx=−(
n
n2−m
)sinmxcosnx+(
m
n2−m
)cosmxsinnx+C
=−2(
n
n2−m
)sinmπcosnπ+2(
m
n2−m
)cosmπsinnπ
Now ... I figure that if n=m, then I can just as well replace them all with a 3rd variable... say z...
=−2(
z
z2−z
)sinzπcoszπ+2(
z
z2−z
)coszπsin
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