May i know how to integrate [1 / (4 + x^2)^1/2] using trig?

[teng125]

may i know how to integrate [1 / (4 + x^2)^1/2] using trigo substitution??

pls help...

 

[TD]

Well, you have a few trig identities, see which one is useful.
Here, you could use the fact that $\sec ^2 x = 1 + \tan ^2 x$.

So try the substitution $x = 2\tan y \Leftrightarrow dx = \frac{2}{{\cos ^2 y}}dy$

 

[teng125]

oh,i got it...can u pls tell me how do you know which frigo identities to choose when u you see the question in general??(not the question above)

pls...

 

[TD]

For trigonometric substitutions, there are two fundamental identities which are often used:

$$\begin{array}{l} \cos ^2 x + \sin ^2 x = 1 \Leftrightarrow \cos ^2 x = 1 - \sin ^2 x \\ \sec ^2 x = 1 + \tan ^2 x \Leftrightarrow \tan ^2 x = \sec ^2 x - 1 \\ \end{array}$$

The first one can be used for radical expressions of the form $\sqrt {a^2 - x^2 }$ where you then choose the substitution $x = a\sin y$.
The second one can be used for two types: yours, which was of the form $\sqrt {x^2 + a^2 }$ (you then do $x = a\tan y$) or those of the form $\sqrt {x^2 - a^2 }$ (then it's $x = a\sec y$).