int_{pi /2}^{pi} frac{1 - sin x}{1 - cos x} , | vedclass

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(A) ધારો કે $I = \int_{\pi /2}^{\pi} \frac{1 - \sin x}{1 - \cos x} \, dx$. નિત્યસમ $1 - \cos x = 2 \sin^2(x/2)$ અને $\sin x = 2 \sin(x/2) \cos(x/2)$ ન

Vedclass · Accepted Answer

$1 - \ln 2$

Vedclass · Answer

(A) ધારો કે $I = \int_{\pi /2}^{\pi} \frac{1 - \sin x}{1 - \cos x} \, dx$. નિત્યસમ $1 - \cos x = 2 \sin^2(x/2)$ અને $\sin x = 2 \sin(x/2) \cos(x/2)$ નો ઉપયોગ કરતા: $I = \int_{\pi /2}^{\pi} \frac{1 - 2 \sin(x/2) \cos(x/2)}{2 \sin^2(x/2)} \, dx$ $I = \int_{\pi /2}^{\pi} \left( \frac{1}{2} \csc^2(x/2) - \frac{2 \sin(x/2) \cos(x/2)}{2 \sin^2(x/2)} \right) \, dx$ $I = \int_{\pi /2}^{\pi} \left( \frac{1}{2} \csc^2(x/2) - \cot(x/2) \right) \, dx$ પદવાર સંકલન કરતા: $\int \frac{1}{2} \csc^2(x/2) \, dx = \frac{1}{2} \cdot (-2 \cot(x/2)) = -\cot(x/2)$ $\int -\cot(x/2) \, dx = -2 \ln|\sin(x/2)|$ તેથી,$I = [-\cot(x/2) - 2 \ln|\sin(x/2)|]_{\pi /2}^{\pi}$ સીમાઓ મૂકતા: $x = \pi$ માટે: $-\cot(\pi/2) - 2 \ln(\sin(\pi/2)) = 0 - 2 \ln(1) = 0$ $x = \pi/2$ માટે: $-\cot(\pi/4) - 2 \ln(\sin(\pi/4)) = -1 - 2 \ln(1/\sqrt{2}) = -1 - 2 \ln(2^{-1/2}) = -1 - 2(-1/2) \ln 2 = -1 + \ln 2$ $I = 0 - (-1 + \ln 2) = 1 - \ln 2$.

$\int_{\pi /2}^{\pi} \frac{1 - \sin x}{1 - \cos x} \, dx =$

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