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Question

If I=π/20sin3xcosxsin4x+cos4xdx

then I is equal to?

A
π/8
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B
π/4
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C
π/2
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D
π
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Solution

The correct option is A π/8
Using a0f(x)dx=a0f(ax)dx we get
I=π/20cos3xsinxcos4x+sin4xdx(2)
Adding (1) and (2), we get
2I=π/20sinxcosxcos4x+sin4xdx
Using cos4x+sin4x=(cos2x)2+(sin2x)2
=(1/4)(1 cos2x)2+(1/4)(1+cos2x)2
=(1/2)(1+cos22x)
2I=π/202sinxcosx1+cos22xdx
Substitute cos2x=t, so that
2I=1211dt1+t2=10dt1+t2=π4

I=π/8

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