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How do you integrate $$\int_1^{e }\sqrt{t^2+\frac{1}{t^2}+2} \, \mathrm{d}t \, \, \, ?$$

Zain Patel
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user1068636
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  • As a general piece of advice, when you have a sum of powers of a single variable, it's a good habit to write them in order of decreasing (or increasing) exponent. Here, that means writing your radicand as $t^{2} + 2 + \dfrac{1}{t^{2}}$, which may be likelier to suggest the helpful factorization. – Andrew D. Hwang Sep 17 '16 at 19:45
  • Thank you. Does anybody know of a place I can find similar factoring integration problems like this one? – user1068636 Sep 18 '16 at 00:07
  • What I meant is if someone could please point me to a website with integration problems that require some manipulation before the solution becomes obvious. – user1068636 Sep 18 '16 at 00:08

2 Answers2

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Hint: $$\left(t + \frac{1}{t}\right)^2 = t^2 + 2 + \frac{1}{t^2}$$

Zain Patel
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Hint: $$t^2+\frac{1}{t^2}+2 = t^2 + \frac{1}{t^2}+2t\frac{1}{t} = \left(t+\frac{1}{t}\right)^2,$$and you certainly can compute $$\int_1^e t+\frac{1}{t}\,{\rm d}t.$$

Ivo Terek
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    It is important to note, that for $t\in(1,e)\subset\mathbb{R}_+$, $t+1/t > 0$. Because in general, $\sqrt{x^2} = |x|$. –  Sep 17 '16 at 19:29
  • I myself got tangled up in this for a few minutes - the interested reader can consult this Q&A: http://math.stackexchange.com/questions/547792/square-root-of-x2 –  Sep 17 '16 at 19:35