Question

Integrate the following

$\frac{1}{x( {x}^{3} + 1)}$

Please tell the complete steps
​

Answer 1

Step-by-step explanation:

yan na po pic pa heart po

pa brainlest. po thank you po god blesssss ingat stay safe po

Answer 2

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

Given integral is

$\rm :\longmapsto\:\displaystyle\int\rm \frac{dx}{x( {x}^{3} + 1)}$

can be rewritten as

$\rm \:  =  \: \displaystyle\int\rm \frac{dx}{x \times {x}^{3} \bigg[1 + \dfrac{1}{ {x}^{3} } \bigg]}$

$\rm \:  =  \: \displaystyle\int\rm \frac{dx}{ {x}^{4} (1 + {x}^{ - 3} )}$

Now, To evaluate this integral, we have to use Method of Substitution.

So, Substitute

$\purple{\rm :\longmapsto\:1 + {x}^{ - 3} = y}$

$\purple{\rm :\longmapsto\: - 3{x}^{ - 4}dx = dy}$

$\purple{\rm :\longmapsto\:\dfrac{dx}{ {x}^{4} } = - \dfrac{dy}{3}}$

So, on substituting these values, we get

$\rm \:  =  \: - \dfrac{1}{3}\displaystyle\int\rm \frac{dy}{y}$

$\rm \:  =  \: - \dfrac{1}{3}log |y| + c$

$\rm \:  =  \: - \dfrac{1}{3}log |1 + {x}^{ - 3} | + c$

$\rm \:  =  \: - \dfrac{1}{3}log \bigg|1 + \dfrac{1}{ {x}^{3} } \bigg| + c$

$\rm \:  =  \: - \dfrac{1}{3}log \bigg| \dfrac{ {x}^{3} + 1}{ {x}^{3} } \bigg| + c$

OR

$\rm \:  =  \: \dfrac{1}{3}log \bigg| \dfrac{ {x}^{3}}{ {x}^{3} + 1} \bigg| + c$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

LEARN MORE

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$