A) \[\frac{12}{13}x-\frac{5}{13}\log (3\cos x+2\sin x)\]
B) \[\frac{12}{13}x+\frac{5}{13}\log (3\cos x+2\sin x)\]
C) \[\frac{13}{12}x+\frac{5}{13}\log (3\cos x+2\sin x)\]
D) None of these
Correct Answer: A
Solution :
Write \[{{N}^{r}}=l({{D}^{r}})+m\] (differential coefficient of \[{{D}^{r}}).\] Let 3sinx+2cosx = l(3cosx+2sinx)+m(?3sinx + 2cosx) Comparing coefficients of \[\sin x\] and \[\cos x\]on both sides \[3=2l-3m\] and \[2=3l+2m\] Solving, we get \[l=\frac{12}{13},\] \[m=-\frac{5}{13},\] \[\therefore \,\,I=l\,\int_{{}}^{{}}{dx}+m\int_{{}}^{{}}{\frac{-3\sin x+2\cos x}{3\cos x+2\sin x}\,dx}\] =lx+mlog(3cosx+2sinx)\[=\frac{12}{13}x-\frac{5}{13}\]log(3cosx+2sinx).
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