Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\displaystyle\int{ \frac{ 1 }{ ( { x }^{ 2 } -4) \sqrt{ x+1 } } }d x$$

Answer

(d*x)/((x+2)*(x-2)*sqrt(x+1))

Solution


Rewrite \({x}^{2}-4\) in the form \({a}^{2}-{b}^{2}\), where \(a=x\) and \(b=2\).
\[\int 1 \, \frac{d}{({x}^{2}-{2}^{2})\sqrt{x+1}}dx\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\int 1 \, \frac{d}{(x+2)(x-2)\sqrt{x+1}}dx\]
Use this rule: \(\int a \, dx=ax+C\).
\[\frac{dx}{(x+2)(x-2)\sqrt{x+1}}\]
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