Advertisements
Advertisements
Question
Integrate the function `1/sqrt((x - a)(x - b))`
Advertisements
Solution
Let `I = int dx/sqrt((x - a) (x - b))`
`= int dx/sqrt(x^2 - (a + b)x + ab)`
`= int dx/{[x^2 - 2 ((a + b)/2) x + ((a + b)/2)^2] + ab - ((a + b)/2)^2`
`= int dx/sqrt((x - (a + b)/2)^2 - ((a - b)/2)^2)`
`= log [(x - (a + b)/2) + sqrt((x - (a + b)/2)^2 - ((a - b)/2)^2)] + C` `...[∵ int dx/ sqrt (x^2 - a^2) = log |x + sqrt (x^2 - a^2)| + C]`
`= log [x - (a + b)/2 + sqrt((x - a)(x - b))] + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`
Find:
`int(x^3-1)/(x^3+x)dx`
Integrate the function `1/sqrt(9 - 25x^2)`
Integrate the function `(3x)/(1+ 2x^4)`
Integrate the function `x^2/(1 - x^6)`
Integrate the function `(sec^2 x)/sqrt(tan^2 x + 4)`
Integrate the function `1/sqrt(7 - 6x - x^2)`
Integrate the function `1/sqrt(8+3x - x^2)`
Integrate the function `(4x+ 1)/sqrt(2x^2 + x - 3)`
Integrate the function `(6x + 7)/sqrt((x - 5)(x - 4))`
Integrate the function `(x + 3)/(x^2 - 2x - 5)`
Integrate the function:
`sqrt(1- 4x^2)`
Integrate the function:
`sqrt(x^2 + 4x +1)`
Integrate the function:
`sqrt(x^2 + 3x)`
Integrate the function:
`sqrt(1+ x^2/9)`
`int sqrt(1+ x^2) dx` is equal to ______.
Evaluate : `int_2^3 3^x dx`
\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]
Find:
`int_(-pi/4)^0 (1+tan"x")/(1-tan"x") "dx"`
Find: `int (dx)/(x^2 - 6x + 13)`
`int (a^x - b^x)^2/(a^xb^x)dx` equals ______.
Evaluate \[\int \frac{dx}{x^2-a^2}\].
Evaluate \[\int \frac{dx}{x^2+a^2}\].
Evaluate \[\int \frac{dx}{\sqrt{x^2+a^2}}\].
Which trigonometric substitution is suitable for \[x^2+a^2\] and \[\sqrt{x^2+a^2}\]?
Which method transforms \[\int\frac{dx}{ax^2+bx+c}\] or \[\int\frac{dx}{\sqrt{ax^2+bx+c}}\] into an expression compatible with standard formulae?
For \[\int\frac{px+q}{ax^2+bx+c}\,dx\], how is \[px+q\] written in relation to the derivative of the denominator?
What is the completed-square form of \[x^2-6x+13\]?
Evaluate \[\int\frac{dx}{x^2-6x+13}\].
What is the completed-square form of \[3x^2+13x-10\]?
Evaluate \[\int\frac{dx}{3x^2+13x-10}\].
Which is the correct split for \[\int\frac{x+2}{2x^2+6x+5}\,dx\]?
Which statement is correct when integrating an expression that is not immediately a standard integral?
