નિશ્ચિત સંકલન int_{frac{pi}{6}}^{frac{pi}{3}} | vedclass

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Question

(C) ધારો કે $I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\sqrt{	an x}}$.ગુણધર્મ $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ નો ઉપયોગ ક

Vedclass · Accepted Answer

$\frac{\pi}{12}$

Vedclass · Answer

(C) ધારો કે $I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\sqrt{	an x}}$.ગુણધર્મ $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ નો ઉપયોગ કરતા,જ્યાં $a = \frac{\pi}{6}$ અને $b = \frac{\pi}{3}$,આપણને $a+b = \frac{\pi}{2}$ મળે છે.$I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\sqrt{	an(\frac{\pi}{2}-x)}} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\sqrt{\cot x}} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\frac{1}{\sqrt{	an x}}} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{	an x}}{\sqrt{	an x}+1} dx$.$I$ માટેના બંને પદોનો સરવાળો કરતા:$2I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left( \frac{1}{1+\sqrt{	an x}} + \frac{\sqrt{	an x}}{1+\sqrt{	an x}} ight) dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 dx$.$2I = [x]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}$.તેથી,$I = \frac{\pi}{12}$.

નિશ્ચિત સંકલન $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}$ ની કિંમત શોધો.

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