int_0^pi frac{1}{1+sin x} dx का मान ज्ञात कीज | vedclass

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(B) माना $I = \int_0^\pi \frac{1}{1+\sin x} dx$. सर्वसमिका $\sin x = \frac{2 	an(x/2)}{1+	an^2(x/2)}$ का उपयोग करने पर: $I = \int_0^\pi \frac{1}{1+\

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$2$

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(B) माना $I = \int_0^\pi \frac{1}{1+\sin x} dx$. सर्वसमिका $\sin x = \frac{2 	an(x/2)}{1+	an^2(x/2)}$ का उपयोग करने पर: $I = \int_0^\pi \frac{1}{1+\frac{2 	an(x/2)}{1+	an^2(x/2)}} dx = \int_0^\pi \frac{1+	an^2(x/2)}{1+	an^2(x/2)+2 	an(x/2)} dx$. चूंकि $1+	an^2(x/2) = \sec^2(x/2)$,इसलिए: $I = \int_0^\pi \frac{\sec^2(x/2)}{(1+	an(x/2))^2} dx$. माना $t = 	an(x/2)$,तो $dt = \frac{1}{2} \sec^2(x/2) dx$,जिसका अर्थ है $\sec^2(x/2) dx = 2 dt$. जब $x 	o 0$,तब $t 	o 0$. जब $x 	o \pi$,तब $t 	o \infty$. अतः,$I = \int_0^\infty \frac{2 dt}{(1+t)^2} = 2 \left[ -\frac{1}{1+t} ight]_0^\infty = 2(0 - (-1)) = 2$.

$\int_0^\pi \frac{1}{1+\sin x} dx$ का मान ज्ञात कीजिए।

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