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$$\int^2_{-2}\frac{x^2}{1+5^x}$$

How do I start to integrate this?

I know the basics and tried substituting $5^x$ by $u$ where by changing the base of logarithm I get $\frac{\ln(u)}{\ln 5}=x$, but I got stuck.

Any hints would suffice preferably in the original question and not after my substitution.

(And also using the basic definite integrals property.)

Now I know only basic integration, that is restricted to high school, so would prefer answer in terms of that level.

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  • $\begingroup$ @PabloRotondo I might not know that property ,the only properties I do know are in page 55 of this link [ncert.nic.in/ncerts/l/lemh201.pdf] $\endgroup$ Mar 14, 2016 at 9:32
  • $\begingroup$ You'll spend a hard time trying to find an antiderivative. Follow @GoodDeeds. $\endgroup$
    – user65203
    Mar 14, 2016 at 9:39
  • $\begingroup$ See also this question. $\endgroup$ Jan 6, 2017 at 8:09
  • $\begingroup$ @MartinSleziak Oh you were linking that question asked today to my question asked 9 months ago. $\endgroup$ Jan 6, 2017 at 17:52
  • $\begingroup$ Yes, I know. I think that the answers to the question might be useful to people reading your post, so I added a link. (The answers there are a bit more detailed.) $\endgroup$ Jan 6, 2017 at 18:15

2 Answers 2

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$$\tag1I=\int_{-2}^{2}\frac{x^2}{1+5^x}dx$$ Note that $$\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$$ Thus, $$\tag2I=\int_{-2}^{2}\frac{(-2+2-x)^2}{1+5^{-2+2-x}}dx=\int_{-2}^{2}\frac{x^2}{1+5^{-x}}dx=\int_{-2}^{2}\frac{5^xx^2}{1+5^{x}}dx$$

Add $(1)$ and $(2)$.

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  • $\begingroup$ So after this I would be left with $x^3$ Right?(for the variable part before appplying upper and lower limit) $\endgroup$ Mar 14, 2016 at 9:40
  • $\begingroup$ @IshanTaneja No, you will be left with $\frac{x^2}{2}$ in the integral $\endgroup$
    – GoodDeeds
    Mar 14, 2016 at 9:40
  • $\begingroup$ I guess that @IshanTaneja means $\frac{x^3}6$, after integration. $\endgroup$
    – user65203
    Mar 14, 2016 at 9:43
  • 1
    $\begingroup$ I always get a grin on my face, when I see someone using this kind of "trick" . Even though it is actually so obvious, I rarely see people using it. It's so wonderful :) $\endgroup$
    – Imago
    Mar 14, 2016 at 9:53
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Hint:

$$\frac1{1+5^{-x}} + \frac1{1+5^x} = 1$$

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  • $\begingroup$ But how do I replace x by -x without any particular property in mind? $\endgroup$ Mar 14, 2016 at 9:34
  • $\begingroup$ @IshanTaneja: Break the integral up into two pieces, one from $[-2,0]$ and one from $[0,2]$. Sub $x \mapsto -x$ in the former integral. $\endgroup$
    – Ron Gordon
    Mar 14, 2016 at 9:36
  • $\begingroup$ Okay I got limit breaking but substituing x by -x will change dx by -dx ,right? $\endgroup$ Mar 14, 2016 at 9:37
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    $\begingroup$ @IshanTaneja: yes, but that negative will disappear when you reverse the upper and lower limits of that integral. $\endgroup$
    – Ron Gordon
    Mar 14, 2016 at 9:38

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