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Question

If 3cosx+2sinx4sinx+5cosxdx=ax+bln|4sinx+5cosx|+C, then which of the following is/are correct?
(where a,b are fixed constants and C is integration constant)

A
a+b=1541
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B
a2b=1941
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C
ab=461681
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D
ab=223
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Solution

The correct option is C ab=461681
Let
3cosx+2sinx=A(4sinx+5cosx)+Bddx(4sinx+5cosx)=A(4sinx+5cosx)+B(4cosx5sinx)
Comparing the coefficients of sinx and cosx, we get
4A5B=25A+4B=3
Solving, we get
A=2341 and B=241
Thus the given integral reduces to
I=2341dx+2414cosx5sinx4sinx+5cosxdx =2341x+241ln|4sinx+5cosx|+C
Thus,
a2b=1941ab=461681

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