integrate 1/(x(x ^ 2 + 4)) dx​

Answer:

[tex]\qquad\boxed{ \sf{ \:\bf \:\displaystyle\int\bf \dfrac{dx}{x( {x}^{2} + 4)} = \dfrac{1}{8}log \bigg| \dfrac{ {x}^{2}}{ {x}^{2} + 4} \bigg | + c \: }}\\ \\ [/tex]

Step-by-step explanation:

Given integral is

[tex]\sf \:\displaystyle\int\sf \dfrac{dx}{x( {x}^{2} + 4)} \\ \\ [/tex]

[tex]\sf \: = \: \displaystyle\int\sf \dfrac{dx}{x \times {x}^{2} \left(1 + \dfrac{4}{ {x}^{2} } \right)} \\ \\ [/tex]

[tex]\sf \: = \: \displaystyle\int\sf \dfrac{dx}{{x}^{3} \left(1 + \dfrac{4}{ {x}^{2} } \right)} \\ \\ [/tex]

[tex]\sf \: = \: \displaystyle\int\sf \dfrac{ {x}^{ - 3} }{1 + {4x}^{ - 2} } \: dx \\ \\ [/tex]

Now, Substitute

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

[tex]\sf \:0 - {8x}^{ - 2 - 1} = \dfrac{dy}{dx} \\ \\ [/tex]

[tex]\sf\implies \sf \:{x}^{ - 3} \: dx \: = \: \dfrac{dy}{ - 8} \\ \\ [/tex]

So, on substituting these values in above integral, we get

[tex]\sf \: = \: - \dfrac{1}{8}\displaystyle\int\sf \dfrac{dy}{y} \\ \\ [/tex]

[tex]\sf \: = \: - \dfrac{1}{8}log |y| + c \\ \\ [/tex]

[tex]\sf \: = \: - \dfrac{1}{8}log |1 + {4x}^{ - 2} | + c \\ \\ [/tex]

[tex]\sf \: = \: - \dfrac{1}{8}log \bigg|1 + \dfrac{4}{ {x}^{2} } \bigg | + c \\ \\ [/tex]

[tex]\sf \: = \: - \dfrac{1}{8}log \bigg| \dfrac{ {x}^{2} + 4}{ {x}^{2} } \bigg | + c \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{8}log \bigg| \dfrac{ {x}^{2}}{ {x}^{2} + 4} \bigg | + c \\ \\ [/tex]

Hence,

[tex]\sf\implies \boxed{ \sf{ \:\sf \:\displaystyle\int\sf \dfrac{dx}{x( {x}^{2} + 4)} = \dfrac{1}{8}log \bigg| \dfrac{ {x}^{2}}{ {x}^{2} + 4} \bigg | + c \: }}\\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {x}^{n} & \sf {nx}^{n - 1}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}[/tex]