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

[srk999]

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.

 

[disregardthat]

For the first: try the substitution $$x = a \sinh(\theta) = a\frac{e^{\theta}-e^{-\theta}}{2}$$

 

[HallsofIvy]

Or use a trig substitution: $x= a tan(\theta)$. Then $a^2+ x^2= a^2+ a^2tan^2(\theta)$$= a^2(1+ tan^2(\theta))= a^2 sec^2(\theta)$. [itex]\sqrt{a^2+ x^2}= |a sec(x)|.

And, of course, $dx= a(tan(\theta))' d\theta= a sec^2(\theta)d\theta$

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