Integrate the function: \(\frac {1}{\sqrt {8+3x-x^2}}\)
Solution and Explanation
8+3x-x2 can be written as 8 - (x2 - 3x + \(\frac 94\) - \(\frac 94\)).
Therefore,
8 - (x2- 3x + \(\frac 94\) - \(\frac 94\))
= \(\frac {41}{4}\) - (x-\(\frac 32\))2
⇒ \(∫\)\(\frac {1}{\sqrt {8+3x-x^2}}\ dx\) = \(∫\frac {1}{\sqrt {\frac {41}{4}-(x-\frac 32)^2}} dx\)
Let x-\(\frac 32\) = t
∴ dx = dt
⇒ \(∫\frac {1}{\sqrt {\frac {41}{4}-(x-\frac 32)^2}}\ dx\) = \(∫\frac {1}{\sqrt {(\frac {\sqrt {41}}{2})^2-t^2}} \ dt\)
= \(sin^{-1}(\frac {t}{\frac {\sqrt {41}}{2}})+ C\)
= \(sin^{-1}(\frac {x-\frac 32}{\frac {\sqrt {41}}{2}})+ C\)
= \(sin^{-1}(\frac {2x-3}{\sqrt {41}})+C\)
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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.
