How to solve these integrals? sqrt(a^2 + x^2) & sqrt(2 + x^2)?

  • Thread starter srk999
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How to solve these integrals, such as- sqrt(a^2 + x^2) & sqrt(2 + x^2)

Please be as descriptive and simple as possible.

Please use only sin and cos if possible We are not allowed a calculator in the exam and will have to find numerical values.
 

Answers and Replies

  • #2
disregardthat
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For the first: try the substitution [tex]x = a \sinh(\theta) = a\frac{e^{\theta}-e^{-\theta}}{2}[/tex]
 
  • #3
HallsofIvy
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Or use a trig substitution: [itex]x= a tan(\theta)[/itex]. Then [itex]a^2+ x^2= a^2+ a^2tan^2(\theta)[/itex][itex]= a^2(1+ tan^2(\theta))= a^2 sec^2(\theta)[/itex]. [itex]\sqrt{a^2+ x^2}= |a sec(x)|.

And, of course, [itex]dx= a(tan(\theta))' d\theta= a sec^2(\theta)d\theta[/itex]

I hope you recognize that the second problem, with [itex]\sqrt{2+ x^2}[/itex], is exactly the same as the first problem with [itex]a= \sqrt{2}[/itiex].
 

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