Integrate the function: \(\frac {1}{\sqrt {(x-1)(x-2)}}\)
Solution and Explanation
(x-1)(x-2) can be written as x2-3x+2.
Therefore,
x2 - 3x + 2
= x2 - 3x +\(\frac 94\) -\(\frac 94\) + 2
=(x - \(\frac 32\))2 - \(\frac 14\)
=(x - \(\frac 32\))2 - (\(\frac 12\))2
∴ \(∫\)\(\frac {1}{\sqrt {(x-1)(x-2)}}\ dx\)= \(∫\frac {1}{\sqrt {(x-\frac 32)^2-(\frac 12)^2}} dx\)
Let x-\(\frac 32\) = t
∴ dx = dt
⇒ \(∫\frac {1}{\sqrt {(x-\frac 32)^2-(\frac 12)^2}} dx\) = \(∫\frac {1}{\sqrt {t^2-(\frac 12)^2}} dt\)
= \(log\ |t+\sqrt {t^2-(\frac 12)^2}|+C\)
=\(log\ |(x-\frac 32)+\sqrt {x^2-3x+2}|+C\)
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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.
