integrate the following
Share
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
Question :- Evaluate the following integral
[tex]\sf \: \displaystyle\int\sf \frac{1}{ \sqrt{x - 5} - \sqrt{x + 3} } \: dx \\ \\ [/tex]
[tex] \\ \large\underline{\sf{Solution-}}[/tex]
Given integral is
[tex]\sf \: \displaystyle\int\sf \frac{1}{ \sqrt{x - 5} - \sqrt{x + 3} } \: dx \\ \\ [/tex]
On rationalizing the denominator, we get
[tex]\sf \: = \: \displaystyle\int\sf \frac{1}{ \sqrt{x - 5} - \sqrt{x + 3} } \times \frac{ \sqrt{x - 5} + \sqrt{x + 3} }{ \sqrt{x - 5} + \sqrt{x + 3} } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf \frac{ \sqrt{x - 5} + \sqrt{x + 3} }{ (\sqrt{x - 5})^{2} - (\sqrt{x + 3})^{2} } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf \frac{ \sqrt{x - 5} + \sqrt{x + 3} }{ (x - 5) - (x + 3) } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf \frac{ \sqrt{x - 5} + \sqrt{x + 3} }{x - 5 - x - 3 } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf \frac{ \sqrt{x - 5} + \sqrt{x + 3} }{ - 8} \: dx \\ \\ [/tex]
[tex]\sf \: = - \dfrac{1}{8}\displaystyle\int\sf ( \sqrt{x - 5} + \sqrt{x + 3} ) \: dx \\ \\ [/tex]
[tex]\sf \: = - \dfrac{1}{8}\bigg[ \dfrac{ {\bigg(x - 5\bigg) }^{\dfrac{3}{2} } }{\dfrac{3}{2} } \:+\dfrac{ {\bigg(x + 3\bigg) }^{\dfrac{3}{2} } }{\dfrac{3}{2} }\: \bigg] \: + c \\ \\ [/tex]
[tex]\sf \: = - \dfrac{1}{12}\bigg[ {\bigg(x - 5\bigg) }^{\dfrac{3}{2} } \: + {\bigg(x + 3\bigg) }^{\dfrac{3}{2} } \bigg] \: + c \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]
Answer:
hope you understand it .