int e^{x} left[ frac{1+sin x}{1+cos x} right] | vedclass

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(D) ધારો કે $I = \int e^{x} \left[ \frac{1+\sin x}{1+\cos x} \right] dx$. નિત્યસમ $1+\cos x = 2\cos^2 \frac{x}{2}$ અને $\sin x = 2\sin \frac{x}{2} \co

Vedclass · Accepted Answer

$e^{x} 	an \frac{x}{2} + C$

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(D) ધારો કે $I = \int e^{x} \left[ \frac{1+\sin x}{1+\cos x} ight] dx$. નિત્યસમ $1+\cos x = 2\cos^2 \frac{x}{2}$ અને $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}$ નો ઉપયોગ કરતા: $I = \int e^{x} \left[ \frac{1 + 2\sin \frac{x}{2} \cos \frac{x}{2}}{2\cos^2 \frac{x}{2}} ight] dx$ $I = \int e^{x} \left[ \frac{1}{2\cos^2 \frac{x}{2}} + \frac{2\sin \frac{x}{2} \cos \frac{x}{2}}{2\cos^2 \frac{x}{2}} ight] dx$ $I = \int e^{x} \left[ \frac{1}{2} \sec^2 \frac{x}{2} + 	an \frac{x}{2} ight] dx$. આપણે જાણીએ છીએ કે $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. અહીં,$f(x) = 	an \frac{x}{2}$ લેતા,$f'(x) = \sec^2 \frac{x}{2} \cdot \frac{1}{2} = \frac{1}{2} \sec^2 \frac{x}{2}$ મળે. તેથી,સંકલનનું મૂલ્ય $e^{x} 	an \frac{x}{2} + C$ થાય.

$\int e^{x} \left[ \frac{1+\sin x}{1+\cos x} \right] dx$ નું મૂલ્ય શું છે?

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