int_0^{pi /2} {log left( {frac{{4 + 3sin x}}{ | vedclass

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Question

(C) ધારો કે $I = \int_0^{\pi /2} {\log \left( {\frac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right)} \,dx$. ગુણધર્મ $\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx

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Vedclass · Answer

(C) ધારો કે $I = \int_0^{\pi /2} {\log \left( {\frac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right)} \,dx$. ગુણધર્મ $\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx$ નો ઉપયોગ કરતા: $I = \int_0^{\pi /2} {\log \left( {\frac{{4 + 3\sin(\pi/2 - x)}}{{4 + 3\cos(\pi/2 - x)}}} \right)} \,dx$. કારણ કે $\sin(\pi/2 - x) = \cos x$ અને $\cos(\pi/2 - x) = \sin x$,તેથી: $I = \int_0^{\pi /2} {\log \left( {\frac{{4 + 3\cos x}}{{4 + 3\sin x}}} \right)} \,dx$. ગુણધર્મ $\log(a/b) = -\log(b/a)$ નો ઉપયોગ કરતા: $I = \int_0^{\pi /2} -\log \left( {\frac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right) \,dx = -I$. તેથી,$2I = 0$,જેનો અર્થ છે કે $I = 0$.

$\int_0^{\pi /2} {\log \left( {\frac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right)} \,dx$ નું મૂલ્ય શોધો.

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