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Question

If the relation between the order of integrals of sin(x) can be given by
sinn (x) dx=sinn1(x).cos(x)n+n1n sinn2 (x) dx; n > 0
Then find sin3 (x) dx.

A
cos(x)3(1+sin2(x))+C
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B
cos(x)5(1+sin2(x))+C
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C
sin(x)3(1+sin2(x))+C
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D
sin(x)5(1+sin2(x))+C
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Solution

The correct option is A cos(x)3(1+sin2(x))+C
The given relation is nothing but the reduction formula for sin(x). We’ll use the the above formula for n = 3.
So, sin3 (x) dx=sin2(x).cos(x)3+13 sin (x) dx
sin3 (x) dx=sin2(x).cos(x)313 cos(x)+C (Using sin (x) dx=cos(x))
Or cos(x)3(1+sin2(x))+C

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