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Value of integration of (sin^n x cos^m+1 x) from 0 to π is where m,n belong to N

Value of integration of (sin^n x cos^m+1 x) from 0 to π is where m,n belong to N

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Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
5 years ago
hello aditi, i am aditya :)
we know that we can write ∫f(x)dx (from a to b) = ∫f(a+b – x)dx (from a to b)
now here a=0 and b=pi
and f(x)=sin^n(x)*cos^(2m+1)(x)
so f(0+pi – x)= f(pi – x)= sin^n(pi – x)*cos^(2m+1)(pi – x)
but we know that sin(pi – x)=sinx and cos(pi – x)= – cosx, also (-1)^(2m+1)= – 1
so f(pi – x)= – sin^n(x)*cos^(2m+1)(x)= – f(x)
let J= ∫f(x)dx
then J=∫f(pi – x)dx= – ∫f(x)dx
adding, 2J=0 or J=0
so integral is zero.
now look at option c, let K= ∫cos^(2m+1)(x)dx from 0 to pi.
then K= ∫cos^(2m+1)(pi – x)dx from 0 to pi
K= – ∫cos^(2m+1)(x)dx from 0 to pi.
adding, 2K=0 or K=0 
so that J=K=0
hence option C is the correct answer

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