int_{pi /4}^{3pi /4} frac{dx}{1 + cos x} ની ક | vedclass

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(A) ધારો કે $I = \int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$.નિત્યસમ $1 + \cos x = 2\cos^2(x/2)$ નો ઉપયોગ કરતા:$I = \int_{\pi /4}^{3\pi /4} \frac{d

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(A) ધારો કે $I = \int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$.નિત્યસમ $1 + \cos x = 2\cos^2(x/2)$ નો ઉપયોગ કરતા:$I = \int_{\pi /4}^{3\pi /4} \frac{dx}{2\cos^2(x/2)} = \frac{1}{2} \int_{\pi /4}^{3\pi /4} \sec^2(x/2) dx$.$\sec^2(x/2)$ નું સંકલન કરતા:$I = \frac{1}{2} [2 	an(x/2)]_{\pi /4}^{3\pi /4} = [	an(x/2)]_{\pi /4}^{3\pi /4}$.સીમાઓ મૂકતા:$I = 	an(3\pi /8) - 	an(\pi /8)$.$	an(3\pi /8) = \cot(\pi /8)$ હોવાથી:$I = \cot(\pi /8) - 	an(\pi /8) = \frac{\cos(\pi /8)}{\sin(\pi /8)} - \frac{\sin(\pi /8)}{\cos(\pi /8)} = \frac{\cos^2(\pi /8) - \sin^2(\pi /8)}{\sin(\pi /8)\cos(\pi /8)}$.$\cos(2	heta) = \cos^2	heta - \sin^2	heta$ અને $2\sin	heta\cos	heta = \sin(2	heta)$ નો ઉપયોગ કરતા:$I = \frac{\cos(\pi /4)}{\frac{1}{2}\sin(\pi /4)} = 2 \cot(\pi /4) = 2(1) = 2$.

$\int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$ ની કિંમત શોધો.

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