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I started off substituting $u = e^x$ so $du = e^x\, dx$, giving me $$\int \frac{u^8}{u^{16}+36}\,du.$$

So then I substituted $v = u^8$ and $dv = 8u^7du$, which gives me $$\int \frac{1}{v^2 + 36}\,dv.$$ At this point I am unsure what to do.

2 Answers 2

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It's nice to recognize that the denominator is $(e^{8x})^2 + 6^2$. This suggests trying to get the integral into the form $\int \frac{1}{u^2+a^2}\,du$, which can be solved by elementary means.

So, let's try that: setting $w=e^{8x}$, we have $dw = 8e^{8x}\,dx$, so the integral becomes $$\int\frac{e^{8x}\,dx}{(e^{8x})^2 + 36} = \frac{1}{8}\int \frac{dw}{w^2+36}.$$ Now, factor out $6^2$ from the denominator, to get $$\frac{1}{8}\int\frac{dw}{w^2 + 36} = \frac{1}{288}\int\frac{{dw}}{\left(\frac{w}{6}\right)^2 + 1}.$$ Now set $u=\frac{w}{6}$, so $6du = dw$. We have: $$\frac{1}{288}\int\frac{dw}{\left(\frac{w}{6}\right)^2 + 1}= \frac{1}{48}\int\frac{du}{u^2+1}.$$ The last integral should be immediate (Hint: $\frac{1}{1+t^2}$ is the derivative of a standard [inverse] function, so its integral is direct). Then taking $u=\frac{w}{6} = \frac{1}{6}e^{8x}$ gets you the integral; you can always differentiate to check.

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    Thanks, I should have noticed the denominator trick you used.2011-01-31
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    I think in the 1/48 part you meant dw on top. But the u^2 + 1 would give me tan^-1(8e^(8x)) but because of the 1/48 I need to use 6e, right?2011-01-31
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    @Finzz: No, I already changed variables to $u$ at that step. The fraction outside became $\frac{6}{288}$ (since $dw = 6du$), which equals $\frac{1}{48}$. $\int\frac{du}{u^2+1} = \arctan(u)+C = \arctan(\frac{w}{6})+C = \arctan(\frac{e^{8x}}{6}) + C$, and you have the $\frac{1}{48}$ multiplying so you should get $$\frac{1}{48}\arctan\left(\frac{1}{6}e^{8x}\right) + C.$$You can check by differentiation.2011-01-31
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    @Finzz: So I don't know why you think you should get $\arctan(8e^{8x})$, or why you think the $\frac{1}{48}$ **outside** the arctangent means you should "use $6e$".2011-01-31
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Ok. I shall start from the first. Put $t =e^{8x}$ So you have

\begin{align*} \int \frac{e^{8x}}{e^{16x} + 36} \ \text{dx} & = \frac{1}{8}\int \frac{1}{t^{2} + 36} \ \text{dt} \\ &= \frac{1}{8}\int \frac{1}{36 \tan^{2}{\theta} + 36} \times 6 \sec^{2}{\theta} \ d{\theta} \ \Bigl[ \text{ Putting} \ t=6\tan{\theta}\Bigr]\\ &= \frac{6}{8} \int \frac{1}{36 \cdot (1+\tan^{2}{\theta})} \cdot \sec^{2}{\theta} \ d\theta \end{align*}

I am sure you can proceed from here. The substitution which you needed was $\tan{\theta}$. Remember the following:

  • Denominator of the form: $a^{2}- x^{2}$ substitute $x = a\sin{\theta}$ or $a \cos{\theta}$

  • Denominator of the form : $a^{2} + x^{2}$ substitute $x = a\tan{\theta}$.

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    I did not downvote, but as soon as you got to the integral with respect to $t$, you could have changed variables so as to get to the integral of a constant times $(s^2+1)^{-1}$, which everyone should know.2011-01-31
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    @Mariano: Oh, Mariano while integrating generally i substitute with $t$ or with $\theta$, i am sure the OP is mature enough to understand the substitution. He just needs a method by which he can do the integral.2011-01-31
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    But one could argue that the substitution you did in passing from your second integral to the third one actually *complicated* things! :)2011-01-31
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    @Mariano: Ok you mean that one. Ok i shall explain it.2011-01-31
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    @Mariano: Edited. Thanks for pointing out.2011-01-31
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    @Chandru1: Oh, my. Don't you know the integral of $\frac{1}{1+u^2}$ ?2011-01-31
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    @Arturo: What do you want to say.2011-01-31
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    @Chandru1: I said what I wanted to say. Would you really try integrating $\frac{1}{1+u^2}$ by first substituting $u = \tan\theta$ and then using trig identities? *I'm* known for liking to do simple things the hard way, but that is beyond even me.2011-01-31
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    @Arturo: Look each one has their own way of doing things. Now let's not argue on this.2011-01-31
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    @Chandru1: If you want to do things by rote and the hard way instead of trying to recognize simpler ways, be my guest. *I* was not arguing, I was merely marveling.2011-01-31
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    I appreciate your answer, but it's easier to do it the way Arturo did and avoid trig, at least for me. Thanks.2011-01-31