MCQ
$\int_0^{\frac{\pi}{4}} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ = .............
- ✓$3 \pi-50 \log _e 2+20 \log _e 5$
- B$3 \pi-25 \log _e 2+10 \log _e 5$
- C$3 \pi-10 \log _e(2 \sqrt{2})+10 \log _e 5$
- D$3 \pi-30 \log _e 2+20 \log _e 5$
$ 136 \sin \mathrm{x}=\mathrm{A}(3 \sin \mathrm{x}+5 \cos \mathrm{x})+\mathrm{B}(3 \cos \mathrm{x}-5 \sin \mathrm{x}) $
$ 136=3 \mathrm{~A}-5 \mathrm{~B} \quad \ldots(1) $
$ 0=5 \mathrm{~A}+3 \mathrm{~B} \quad \ldots(2) $
$ 3 \mathrm{~B}=-5 \mathrm{~A} \Rightarrow \mathrm{B}=-\frac{5}{3} \mathrm{~A} $
$ 136=3 \mathrm{~A}-5\left(-\frac{5}{3} \mathrm{~A}\right) $
$ 136=3 \mathrm{~A}+\frac{25}{3} \mathrm{~A} $
$ 136=\frac{34 \mathrm{~A}}{3} $
$ \Rightarrow \mathrm{A}=\frac{136 \times 3}{34}=12 $
$ \mathrm{~B}=\frac{-5}{3}(12)=-20$
$ I=\int_0^{\pi / 4} \frac{A(3 \sin x+5 \cos x)}{3 \sin x+5 \cos x}+\int_0^{\pi / 4} \frac{B(3 \cos x-5 \sin x)}{3 \sin x+5 \cos x} $
$ =A(x)_0^{\pi / 4}+B[\ln (3 \sin x+5 \cos x)]_0^{\pi / 4} $
$ =12\left(\frac{\pi}{4}\right)-20 \ln \left(\frac{3}{\sqrt{2}}+\frac{5}{\sqrt{2}}\right)-\ln (0+5) $
$ =3 \pi-20 \ln 4 \sqrt{2}+20 \ln 5 $
$ =3 \pi-20 \times \frac{5}{2} \ln 2+20 \ln 5 $
$ =3 \pi-50 \ln 2+20 \ln 5$
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