સંકલન int_{-2}^{2} frac{|x^{3}+x|}{e^{x|x|}+1 | vedclass

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Question

(B) ધારો કે $f(x) = \frac{|x^{3}+x|}{e^{x|x|}+1}$.આપણે ગુણધર્મ $\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx$ નો ઉપયોગ કરીએ છીએ.અહીં $a = 2$

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(B) ધારો કે $f(x) = \frac{|x^{3}+x|}{e^{x|x|}+1}$.આપણે ગુણધર્મ $\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx$ નો ઉપયોગ કરીએ છીએ.અહીં $a = 2$ છે,તેથી $\int_{-2}^{2} f(x) dx = \int_{0}^{2} (f(x) + f(-x)) dx$.કારણ કે $|x^3+x| = |x(x^2+1)| = |x|(x^2+1)$,$x > 0$ માટે,$|x^3+x| = x(x^2+1) = x^3+x$.વળી,$f(-x) = \frac{|(-x)^3+(-x)|}{e^{-x|-x|}+1} = \frac{|-(x^3+x)|}{e^{-x^2}+1} = \frac{x^3+x}{e^{-x^2}+1}$.આમ,$f(x) + f(-x) = \frac{x^3+x}{e^{x^2}+1} + \frac{x^3+x}{e^{-x^2}+1} = (x^3+x) \left( \frac{1}{e^{x^2}+1} + \frac{e^{x^2}}{1+e^{x^2}} \right) = (x^3+x) \left( \frac{1+e^{x^2}}{1+e^{x^2}} \right) = x^3+x$.તેથી,$\int_{0}^{2} (x^3+x) dx = \left[ \frac{x^4}{4} + \frac{x^2}{2} \right]_{0}^{2} = \left( \frac{16}{4} + \frac{4}{2} \right) - 0 = 4 + 2 = 6$.

સંકલન $\int_{-2}^{2} \frac{|x^{3}+x|}{e^{x|x|}+1} dx$ નું મૂલ્ય કેટલું થાય?

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