Question

integration of 1 / 4x+5x^-11​

Answer 1

$\dfrac{1}{4x+5x^{-11}} = \dfrac{1}{4x+\frac{5}{x^{11}}} = \dfrac{x^{11}}{4x^{11}+5}$

$\sf{Let\: 4x^{12}+5=t \implies 4(12)x^{11} .dx = dt}$

$\sf{Therefore,}\\\\\implies \int \dfrac{x^{11}}{4x^{12}+5} dx$

$\\\\\implies \dfrac{1}{4(12)} \int \dfrac{4(12)x^{11}}{4x^{12}+5} dx$

$\\\\\sf{\implies \dfrac{1}{48}\int\dfrac{1}{t} dt}$

$\\\\\implies\sf{\dfrac{log_{e}|t|}{48} +C  }$

$\\\\\implies\sf{\dfrac{ln|4x^{12}+5|}{48} +C }$