int e^{x}left(frac{1+sin x}{1+cos x}right) d | vedclass

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Question

(A) આપેલ સંકલન $ I = \int e^{x} \left( \frac{1+\sin x}{1+\cos x} \right) dx $ છે.ત્રિકોણમિતીય નિત્યસમ $ 1+\cos x = 2\cos^2 \left( \frac{x}{2} \right)

Vedclass · Accepted Answer

$ e^{x} 	an \left(\frac{x}{2}ight)+C $

Vedclass · Answer

(A) આપેલ સંકલન $ I = \int e^{x} \left( \frac{1+\sin x}{1+\cos x} ight) dx $ છે.ત્રિકોણમિતીય નિત્યસમ $ 1+\cos x = 2\cos^2 \left( \frac{x}{2} ight) $ અને $ \sin x = 2\sin \left( \frac{x}{2} ight) \cos \left( \frac{x}{2} ight) $ નો ઉપયોગ કરતા:$ I = \int e^{x} \left( \frac{1 + 2\sin \left( \frac{x}{2} ight) \cos \left( \frac{x}{2} ight)}{2\cos^2 \left( \frac{x}{2} ight)} ight) dx $$ I = \int e^{x} \left( \frac{1}{2\cos^2 \left( \frac{x}{2} ight)} + \frac{2\sin \left( \frac{x}{2} ight) \cos \left( \frac{x}{2} ight)}{2\cos^2 \left( \frac{x}{2} ight)} ight) dx $$ I = \int e^{x} \left( \frac{1}{2} \sec^2 \left( \frac{x}{2} ight) + 	an \left( \frac{x}{2} ight) ight) dx $આપણે જાણીએ છીએ કે $ \int e^{x} (f(x) + f'(x)) dx = e^{x} f(x) + C $.અહીં,$ f(x) = 	an \left( \frac{x}{2} ight) $ લો. તો $ f'(x) = \sec^2 \left( \frac{x}{2} ight) \cdot \frac{1}{2} = \frac{1}{2} \sec^2 \left( \frac{x}{2} ight) $.તેથી,$ I = e^{x} 	an \left( \frac{x}{2} ight) + C $.

$ \int e^{x}\left(\frac{1+\sin x}{1+\cos x}\right) d x $ ની કિંમત શું છે?

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જો $\int e^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) d x=g(x)+C$ જ્યાં $C$ એ સંકલનનો અચળાંક છે,તો $g \left(\frac{1}{2}\right)$ ની કિંમત શોધો :

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