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

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(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 $I = 0$.

List-$I$	List-$II$
$I. \int_{-1}^1 x\|x\| dx$	$(a) \frac{\pi}{2}$
$II. \int_0^{\pi/2} \left(1 + \log \left(\frac{4+3\sin x}{4+3\cos x}\right)\right) dx$	$(b) \int_0^a 2f(x) dx$
$III. \int_0^a f(x) dx$	$(c) \int_0^a [f(x) + f(-x)] dx$
$IV. \int_{-a}^a f(x) dx$	$(d) 0$
	$(e) \int_0^a f(a-x) dx$

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

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