Question

Solve the given integral

$\int \frac{dx}{ \sqrt{x} \sqrt{2 - x} }$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x} \sqrt{2 - x} }$

can be rewritten as

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x(2 - x)} }$

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{2x - {x}^{2} } }$

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - ({x}^{2} - 2x)} }$

can be further rewritten as

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - ({x}^{2} - 2x + 1 - 1)} }$

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - [{(x - 1)}^{2} - 1]} }$

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{1 - {(x - 1)}^{2}} }$

$\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ {1}^{2} - {(x - 1)}^{2}} }$

We know,

$\boxed{ \rm{ \:\displaystyle \int \rm \: \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } \: = \: {sin}^{ - 1} \frac{x}{a} + c \: \: }}$

So, using this result, we get

$\rm \: = \: {sin}^{ - 1} \dfrac{(x - 1)}{1} + c$

$\rm \: = \: {sin}^{ - 1}(x - 1) + c$

Hence,

$\rm\implies \:\boxed{ \rm{ \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x} \sqrt{2 - x} } = \: {sin}^{ - 1}(x - 1) + c \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ {x}^{2} + {a}^{2} } = \dfrac{1}{a} {tan}^{ - 1} \dfrac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log |x + \sqrt{ {x}^{2} + {a}^{2}} | + c}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-}}Solution−

Given integral is

\begin{gathered}\rm \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x} \sqrt{2 - x} } \\ \end{gathered}∫x2−xdx

can be rewritten as

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x(2 - x)} } \\ \end{gathered}=∫x(2−x)dx

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{2x - {x}^{2} } } \\ \end{gathered}=∫2x−x2dx

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - ({x}^{2} - 2x)} } \\ \end{gathered}=∫−(x2−2x)dx

can be further rewritten as

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - ({x}^{2} - 2x + 1 - 1)} } \\ \end{gathered}=∫−(x2−2x+1−1)dx

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ - [{(x - 1)}^{2} - 1]} } \\ \end{gathered}=∫−[(x−1)2−1]dx

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{1 - {(x - 1)}^{2}} } \\ \end{gathered}=∫1−(x−1)2dx

\begin{gathered}\rm \: = \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{ {1}^{2} - {(x - 1)}^{2}} } \\ \end{gathered}=∫12−(x−1)2dx

We know,

\begin{gathered}\boxed{ \rm{ \:\displaystyle \int \rm \: \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } \: = \: {sin}^{ - 1} \frac{x}{a} + c \: \: }} \\ \end{gathered}∫a2−x2dx=sin−1ax+c

So, using this result, we get

\begin{gathered}\rm \: = \: {sin}^{ - 1} \dfrac{(x - 1)}{1} + c \\ \end{gathered}=sin−11(x−1)+c

\begin{gathered}\rm \: = \: {sin}^{ - 1}(x - 1) + c \\ \end{gathered}=sin−1(x−1)+c

Hence,

\begin{gathered}\rm\implies \:\boxed{ \rm{ \: \displaystyle \int \rm \: \frac{dx}{ \sqrt{x} \sqrt{2 - x} } = \: {sin}^{ - 1}(x - 1) + c \: }} \\ \end{gathered}⟹∫x2−xdx=sin−1(x−1)+c

\rule{190pt}{2pt}

Additional Information :-

\begin{gathered}\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ {x}^{2} + {a}^{2} } = \dfrac{1}{a} {tan}^{ - 1} \dfrac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log |x + \sqrt{ {x}^{2} + {a}^{2}} | + c}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}\end{gathered}MoreFormulaeMoreFormulae★∫x2+a2dx=a1tan−1ax+c★∫x2−a2dx=log∣x+x2−a2∣+c★∫a2−x2dx=sin−1ax+c★∫x2+a2dx=log∣x+x2+a2∣+c