Integrate the function: \(\frac {1}{\sqrt {(x-a)(x-b)}}\)
Solution and Explanation
(x-a)(x-b) can be written as x2 - (a+b)x + ab.
Therefore,
x2- (a+b)x + ab
= x2- (a+b)x + \(\frac {(a+b)^2}{4}\) - \(\frac {(a+b)^2}{4}\) + ab
= [x-(\(\frac {a+b}{2}\))]2 - \(\frac {(a-b)^2}{4}\)
⇒ \(∫\)\(\frac {1}{\sqrt {(x-a)(x-b)}}\ dx\) = \(∫\frac {1}{\sqrt {{x-(\frac {a+b}{2})}^2-(\frac {a-b}{2})^2}} dx\)
Let x - (\(\frac {a+b}{2}\)) = t
∴ dx = dt
⇒ \(∫\frac {1}{\sqrt {{x-(\frac {a+b}{2})}^2-(\frac {a-b}{2})^2}} dx\) = \(∫\frac {1}{\sqrt {t^2-(\frac {a-b}{2})^2}}dt\)
= \(log \ |t+\sqrt {t^2-(\frac {a-b}{2})^2|}+C\)
= \(log \ |{x-(\frac {a+b}{2})}+\sqrt {(x-a)(x-b)}|+C\)
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
