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Question

If I=sinx+sin3xcos2xdx=Pcosx+Qlog|f(x)|+R, then
( Q<0)

A
P=1/2,Q=342
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B
P=1/4,Q=12
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C
f(x)=2cosx+12cosx1
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D
f(x)=2cosx12cosx+1
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Solution

The correct options are
C P=1/2,Q=342
D f(x)=2cosx12cosx+1
Given I=sinx+sin3xcos2xdx=Pcosx+Qlog|f(x)|+R,

I=sinx+sin3xcos2xdx

=sinx(1+sin2x)2cos2x1dx

=sinx(2cos2x)2cos2x1dx

Let cosx=tdt=sinxdx.

I=(2t2)2t21dt=t222t21dt=122t242t21dt

=12dt32dt2t21

=12cosx342log2cosx12cosx+1+C ( applying integration formulas)

by comparing with the given we get

P=12,Q=342,f(x)=2cosx12cosx+1

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