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Evaluate $$ \int_{\ln(0.5)}^{\ln(2)}\left( \frac{\displaystyle\sin x \frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+(xe^{\cos x}\sin x)^2}+ 2\sin(x^2+2)\arctan\left(\frac{x^3}{3}\right) } {\displaystyle 1+e^{-\frac{x^2}{2}}+x^7 \sin(-\pi x)+\frac{12}{11}|x|^{2\pi+1}} \,d x\right) $$ This is my solution, it's correct? First we observe that $\ln (0.5)=\ln \frac{1}{2}=-\ln 2$

therefore, the integral is, said $ f (x) $ the integrand, $\int_{-\ln 2}^{\ln 2} f(x)\,\,dx$ that is, an integral over an interval symmetrical about the origin; without taking roads for the search of all the primitives, and by exploiting the symmetry of the interval, we check if the function is odd, in that case one can immediately conclude that the value of 'integral is $ 0 $, then we have: therefore, the integral is, that $ f (x) $ the integrand,

\begin{align} f(-x)&= \frac{\displaystyle\sin (-x)\frac{\sqrt{\sin^2(\cos (-x))+\pi e^{((-x)^4)}}}{1+((-x)e^{\cos (-x)}\sin (-x))^2}+2\sin((-x)^2+2)\arctan\left(\frac{(-x)^3}{3}\right)}{\displaystyle 1+e^{-\frac{(-x)^2}{2}}+(-x)^7 \sin(-\pi (-x))+\frac{12}{11}|(-x)|^{2\pi+1}}\\ &= \frac{\displaystyle-\sin x\frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+( x e^{\cos x}\sin x )^2}-2\sin(x^2+2)\arctan\left(\frac{ x ^3}{3}\right)}{\displaystyle 1+e^{-\frac{x^2}{2}}+x^7 \sin(-\pi x )+\frac{12}{11}|x|^{2\pi+1}}\\ &= -\frac{\displaystyle \sin x\frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+( x e^{\cos x}\sin x )^2}+2\sin(x^2+2)\arctan\left(\frac{ x ^3}{3}\right)}{\displaystyle 1+e^{-\frac{x^2}{2}}+x^7 \sin(-\pi x )+\frac{12}{11}|x|^{2\pi+1}}\\ &=-f(x) \end{align}

the function is therefore odd and therefore the integral is equal to $ 0 $

Michael Hardy
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DeeJay
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1 Answers1

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Yes everything is fine. ${}{}{}{}{}{}{}{}{}{}{}$

leo
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  • This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – gnometorule Mar 04 '13 at 03:58
  • @gnometorule As far as I can see, the question "_it's correct?_" and yes, indeed it is. This was an old unanswered question. I answered it to remove it from the list of unanswered questions. – leo Mar 04 '13 at 04:03
  • To try to remove it. This answer needs at least one in order to do so – leo Mar 04 '13 at 04:04