int_{-1}^1 left(sqrt{1+x+x^2} - sqrt{1-x+x^2} | vedclass

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(B) ધારો કે $I = \int_{-1}^1 \left(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}\right) dx$. ગુણધર્મ $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ નો ઉપયોગ કરતા,અહીં $a

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$0$

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(B) ધારો કે $I = \int_{-1}^1 \left(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}\right) dx$. ગુણધર્મ $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ નો ઉપયોગ કરતા,અહીં $a+b = -1+1 = 0$ છે. તેથી,$I = \int_{-1}^1 \left(\sqrt{1+(-x)+(-x)^2} - \sqrt{1-(-x)+(-x)^2}\right) dx$. $I = \int_{-1}^1 \left(\sqrt{1-x+x^2} - \sqrt{1+x+x^2}\right) dx$. $I = - \int_{-1}^1 \left(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}\right) dx$. $I = -I$. $2I = 0$,જેનો અર્થ છે કે $I = 0$.

$\int_{-1}^1 \left(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}\right) dx$ નું મૂલ્ય શોધો.

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