Question

integration of 1\√(x-3)*(x-2) dx​

Answer 1

$\sf \: Let \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{(x - 3)(x - 2)} }$

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} - 3x - 2x + 6 } }$

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} - 5x + 6 } }$

• Now, use completing squares method, i.e. add and subtract the square of half of the coefficient of x.

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} - 5x + {\bigg(\dfrac{ - 5}{2} \bigg) }^{2} - {\bigg(\dfrac{ - 5}{2} \bigg)}^{2} + 6} }$

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} - 5x + {\bigg(\dfrac{5}{2} \bigg)}^{2} - \dfrac{25}{4} + 6} }$

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {\bigg(x - \dfrac{5}{2} \bigg)}^{2} - \dfrac{1}{4} } }$

$\sf \: \: I \: = \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {\bigg(x - \dfrac{5}{2} \bigg)}^{2} - {\bigg(\dfrac{1}{2} \bigg)}^{2} } }$

$\boxed{\because \: \: \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} - {a}^{2} }} = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }$

$\sf \: I = log\bigg|x - \dfrac{5}{2} + \sqrt{ {\bigg(x - \dfrac{5}{2} \bigg)}^{2} - {\bigg(\dfrac{1}{2} \bigg)}^{2} } \bigg| + c$

$\sf \: I = log \bigg |x - \dfrac{5}{2} + \sqrt{ {x}^{2} - 5x + \dfrac{25}{4} - \dfrac{1}{4} } \bigg| + c$

$\sf \: I = log \bigg |x - \dfrac{5}{2} + \sqrt{ {x}^{2} - 5x + 6} \bigg| + c$

$\sf \: I = log \bigg |\dfrac{2x - 5 + 2 \sqrt{ {x}^{2} - 5x + 6} }{2} \bigg| + c$

$\sf \: I = log \bigg |2x - 5 + 2 \sqrt{ {x}^{2} - 5x + 6 } \bigg| - log(2) + c$

$\sf \: I = log \bigg |2x - 5 + 2 \sqrt{ {x}^{2} - 5x + 6 } \bigg| + d$

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Additional Information :-

$\boxed{\: \: \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {x}^{2} + {a}^{2} }} = log |x + \sqrt{ {x}^{2} + {a}^{2} } | + c }$

$\boxed{ \: \: \displaystyle\int \tt \: \dfrac{dx}{ {x}^{2} - {a}^{2} } = \dfrac{1}{2} log |\dfrac{x - a}{x + a} | + c }$

$\boxed{ \: \: \displaystyle\int \tt \: \dfrac{dx}{ \sqrt{ {a}^{2} - {x}^{2} }} = {sin}^{ - 1} \bigg(\dfrac{x}{a} \bigg) + c }$

$\boxed{ \: \: \displaystyle\int \tt \: \dfrac{dx}{ {x}^{2} + {a}^{2} } =\dfrac{1}{a} {tan}^{ - 1} \bigg(\dfrac{x}{a} \bigg) + c}$