Evaluate the following integral[tex] \int \: \frac{dx}{x \sqrt{ {x}^{4} - 1 } } \\ [/tex]​

Answer:

[tex]\qquad\boxed{ \sf{ \:\sf \:\displaystyle\int\sf \dfrac{1}{x \sqrt{ {x}^{4} - 1} } \: dx \: = \: \dfrac{1}{2} {tan}^{ - 1} \sqrt{ {x}^{4} - 1 } + c \: }}\\ \\ [/tex]

Step-by-step explanation:

Given integral is

[tex]\sf \:\displaystyle\int\sf \dfrac{1}{x \sqrt{ {x}^{4} - 1} } \: dx \\ \\ [/tex]

can be rewritten as

[tex]\sf \: = \: \displaystyle\int\sf \dfrac{ {4x}^{3} }{ {4x}^{3} \times x \sqrt{ {x}^{4} - 1} } \: dx \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{4} \displaystyle\int\sf \dfrac{ {4x}^{3} }{ {x}^{4} \sqrt{ {x}^{4} - 1} } \: dx \\ \\ [/tex]

Now, to evaluate this integral, we substitute

[tex]\sf \: \sqrt{ {x}^{4} - 1} = y \\ \\ [/tex]

[tex]\sf \: {x}^{4} - 1 = {y}^{2} \\ \\ [/tex]

[tex]\sf\implies \sf \: {x}^{4} = {y}^{2} + 1 \\ \\ [/tex]

[tex]\sf\implies \sf \: 4{x}^{3} \: dx = 2y \: dy \\ \\ [/tex]

So, on substituting the values, we get

[tex]\sf \: = \: \dfrac{1}{4} \displaystyle\int\sf \dfrac{ 2y }{( {y}^{2} + 1) y } \: dy\\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{2} \displaystyle\int\sf \dfrac{1 }{ {y}^{2} + 1 } \: dy\\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{2} {tan}^{ - 1}y + c \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{2} {tan}^{ - 1} \sqrt{ {x}^{4} - 1 } + c \\ \\ [/tex]

Hence,

[tex]\sf\implies \boxed{ \sf{ \:\sf \:\displaystyle\int\sf \dfrac{1}{x \sqrt{ {x}^{4} - 1} } \: dx \: = \: \dfrac{1}{2} {tan}^{ - 1} \sqrt{ {x}^{4} - 1 } + 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]