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
a−2b=1941
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C
ab=461681
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D
ab=223
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Solution
The correct option is Cab=461681 Let 3cosx+2sinx=A(4sinx+5cosx)+Bddx(4sinx+5cosx)=A(4sinx+5cosx)+B(4cosx−5sinx)
Comparing the coefficients of sinx and cosx, we get 4A−5B=25A+4B=3
Solving, we get A=2341 and B=241
Thus the given integral reduces to I=2341∫dx+241∫4cosx−5sinx4sinx+5cosxdx=2341x+241ln|4sinx+5cosx|+C
Thus, a−2b=1941ab=461681