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b)
\(I = \int\limits_0^{\pi /2} {\frac{{3\sin x + 4\cos x}}{{3{{\sin }^2}x + 4{{\cos }^2}x}}dx} = 3\int\limits_0^{\pi /2} {\frac{{\sin {\rm{x}}dx}}{{3{{\sin }^2}x + 4{{\cos }^2}x}} + 4} \int\limits_0^{\pi /2} {\frac{{\cos xdx}}{{3{{\sin }^2}x + 4{{\cos }^2}x}}}=I_1+I_2 \)
Trong đó :
\({I_1} = 3\int\limits_0^{\pi /2} {\frac{{\sin {\rm{x}}dx}}{{3{{\sin }^2}x + 4{{\cos }^2}x}} \underbrace{=}_{t= \cos x}3\int\limits_0^{1}\frac{dt}{3+t^2}= \frac{\pi }{{2\sqrt 3 }}} \)
\({I_2} = 4\int\limits_0^{\pi /2} {\frac{{\cos xdx}}{{3{{\sin }^2}x + 4{{\cos }^2}x}}} \underbrace{=}_{t= \sin x}4\int\limits_0^{1}\frac{dt}{4-t^2} = \ln 3\)
\( \Rightarrow I = {I_1} + {I_2} = \boxed {\dfrac{\pi }{{2\sqrt 3 }} + \ln 3} \).
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