Question

Solve this integral asap

Do not spam ​

Answer 1

$\sf\huge\bold{Solution}$

Step by step explanation:

$\dfrac{1}{ \sqrt{\bigg(x - a\bigg)\bigg(x - b\bigg)} }$

$= \dfrac{1}{ \sqrt{x\bigg(x - b\bigg) - a\bigg(x - b\bigg)} }$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {x}^{2} - bx - ax + ab } } dx$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {x}^{2} - x\bigg(a + b\bigg) + ab } } dx$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {\bigg(x - \dfrac{a + b}{2}\bigg )}^{2} - {\bigg( \dfrac{a + b}{2} \bigg)}^{2} + ab } } dx$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {\bigg(x - \dfrac{a + b}{2}\bigg) }^{2} - \dfrac{ {a}^{2} - {b}^{2} - 2ab + 4ab }{4} } } dx$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {\bigg(x - \dfrac{a + b}{2} \bigg)}^{2} - {\bigg( \dfrac{a - b}{2}\bigg) }^{2} } } dx$

$\sf \longmapsto\displaystyle\int \dfrac{1}{ \sqrt{ {\bigg(x - \dfrac{a + b}{2}\bigg) }^{2} - {\bigg( \dfrac{a - b}{2}\bigg) }^{2} } } dx$

$\sf \longmapsto\displaystyle\int \: log\bigg |x + \sqrt{ {x}^{2} - {a}^{2} }\bigg | + C$

$\sf \longmapsto\displaystyle\int \bigg|x - \dfrac{a + b}{2} + \sqrt{ {\bigg(x - \dfrac{a + b}{2} \bigg)}^{2} - {\bigg( \dfrac{a - b}{2}\bigg )}^{2} }\bigg | + C$

$\sf \longmapsto\displaystyle\int \: log\bigg |x - \dfrac{a + b}{2} + \sqrt{ {x}^{2} + {\bigg( \dfrac{a + b}{2}\bigg) }^{2} - 2\bigg(x\bigg)\bigg( \dfrac{a + b}{2} \bigg) - {\bigg( \dfrac{a - b}{2} \bigg)}^{2} } \bigg| + C$

$\sf \longmapsto\displaystyle\int \: log\bigg |x - \dfrac{a + b}{2} + \sqrt{ {x}^{2} - 2\bigg(x\bigg)\bigg( \dfrac{a + b}{2}\bigg ) + {\bigg( \dfrac{a + b}{2}\bigg) }^{2} - {\bigg( \dfrac{a - b}{2}\bigg )}^{2} } \bigg| + C$

$\sf \longmapsto\displaystyle\int \: log\bigg |x - \dfrac{a + b}{2} \sqrt{ {x}^{2} - x\bigg(a + b\bigg) + \dfrac{ {a}^{2} + {b}^{2} + 2ab}{4} - \dfrac{ {a}^{2} + {b}^{2} - 2ab }{4} } \bigg| + C$

$\sf \longmapsto\displaystyle\int \: log\bigg |x - \dfrac{a + b}{2} + \sqrt{ {x}^{2} - x\bigg(a + b\bigg) + \dfrac{2ab}{4} + \dfrac{2ab}{4} }\bigg | + C$

$\sf \longmapsto\displaystyle\int \: log \bigg|x - \dfrac{a + b}{2} + \sqrt{ {x}^{2} -x\bigg( a + b \bigg)+ \dfrac{4ab}{4} } \bigg | + C$

$\sf \longmapsto\displaystyle\int \: log \bigg|x - \dfrac{a + b}{2} + \sqrt{ {x}^{2} - ax - bx + ab} \bigg| + C$

$\sf \longmapsto\displaystyle\int \: log\bigg |x - \dfrac{a + b}{2} + \sqrt{x\bigg(x - a\bigg) - b\bigg(x - a\bigg)} \bigg | + C$

$\sf= log\bigg |x - \dfrac{a + b}{2} + \sqrt{(x - a)(x - b)} \bigg| + C$

Answer 2

EXPLANATION.

$\sf \: \implies \: \int \dfrac{1}{ \sqrt{(x - a)(x - b)} } dx.$

$\sf \: \implies \: \int \dfrac{1}{ \sqrt{( {x}^{2} - xb - xa + ab) } }dx.$

$\sf \: \implies \: \int \dfrac{1}{ \sqrt{( {x}^{2} - (a + b)x + ab} } dx.$

Multiply and divide by 2 in equation, we get.

$\sf \: \implies \: \int \: \dfrac{1}{ \sqrt{( {x}^{2} - 2(x) \bigg ( \dfrac{a + b}{2} \bigg) + ab} }dx.$

Change equation into perfect square, we get.

$\sf \: \implies \: \int \dfrac{1}{ \sqrt{( {x}^{2} - 2x \bigg( \dfrac{a + b}{2} \bigg) + \bigg( \dfrac{a + b}{2} \bigg) {}^{2} - \bigg( \dfrac{a + b}{2} \bigg) {}^{2} + ab} } dx. \\ \\ \\ \sf \: \implies \: \int \: \dfrac{1}{ \sqrt{ \bigg(x - \dfrac{a + b}{2} \bigg) {}^{2} - \bigg( \dfrac{a + b}{2} \bigg) {}^{2} + ab} } dx. \\ \\ \\ \sf \: \implies \: factorise \: the \: equation \: we \: get \\ \\ \\ \sf \: \implies \: \int \dfrac{1}{ \sqrt{ \bigg(x - \dfrac{a + b}{2} \bigg) {}^{2} - \bigg( \dfrac{ {a}^{2} + {b + 2ab}^{2} }{2} + ab \bigg)} } dx.$

$\sf \: \implies \: \int \: \dfrac{1}{ \sqrt{ \bigg(x - \dfrac{a + b}{2} \bigg) {}^{2} + \bigg( \dfrac{ { - a}^{2} - {b}^{2} - 2ab + 4ab}{4} \bigg)} } dx. \\ \\ \\ \sf \: \implies \: \int \: \dfrac{1}{ \sqrt{ \bigg(x - \dfrac{a + b}{2} \bigg) {}^{2} - \bigg( \dfrac{ {a}^{2} + {b}^{2} - 2ab}{4} \bigg)} } dx. \\ \\ \\ \sf \: \implies \: \int \: \dfrac{1}{ \sqrt{ \bigg(x - \dfrac{a + b}{2} \bigg) {}^{2} - \bigg( \dfrac{a - b}{2} \bigg) {}^{2} } } dx.$

$\sf \: \implies \: as \: we \: know \: that \\ \\ \\ \sf \: \implies \: \int \: \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log \bigg |x + \sqrt{ {x}^{2} - {a}^{2} } \bigg| + c$

$\sf \implies \int log \bigg|x - \dfrac{a + b}{2}+\sqrt{x^{2} - 2(x)\bigg(\dfrac{a + b}{2}\bigg)+ \bigg(\dfrac{a + b}{2} \bigg)^{2} - \bigg(\dfrac{a - b}{2} \bigg)^{2} } \bigg| + c$

$\sf \implies \int log \bigg| x - \dfrac{a + b}{2} + \sqrt{x^{2} - x(a + b) + \dfrac{a^{2}+b^{2} +2ab}{4}- \dfrac{a^{2} + b^{2}-2ab }{4} } \bigg| + c$

$\sf \implies log \bigg| x - \dfrac{a + b}{2} + \sqrt{x^{2} - x(a + b) + \dfrac{2ab}{4} + \dfrac{2ab}{4} } \bigg| + c$

$\sf \implies log \bigg | x - \dfrac{a + b}{2} +\sqrt{x^{2} - x(a + b) +\dfrac{4ab}{4} } \bigg| + c$

$\sf \implies log \bigg| x - \dfrac{a + b}{2} +\sqrt{x^{2} - x(a + b) + ab} \bigg| + c$

$\sf \implies log \bigg| x - \dfrac{a + b}{2} +\sqrt{(x - a)(x - b)} \bigg| + c$