Question

integration x^3-x-2dx/1-x^2
​

Answer 1

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

Given integral is

$\rm \: \displaystyle\int\rm \frac{ {x}^{3} - x - 2}{1 - {x}^{2} } \: dx$

can be rewritten as

$\rm \: = \: - \: \displaystyle\int\rm \frac{ {x}^{3} - x - 2}{{x}^{2} - 1} \: dx$

$\rm \: = \: - \: \displaystyle\int\rm \frac{x( {x}^{2} - 1) - 2}{{x}^{2} - 1} \: dx$

$\rm \: = \: - \displaystyle\int\rm x \: dx \: + \: \displaystyle\int\rm \frac{2}{ {x}^{2} - 1} \: dx$

We know,

$\boxed{ \rm{ \: \displaystyle\int\rm {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: \: }}$

and

$\boxed{ \rm{ \: \displaystyle\int\rm \frac{dx}{ {x}^{2} - {a}^{2} } = \frac{1}{2a}log\bigg |\dfrac{x - a}{x + a} \bigg| + c \: }}$

So, using these results, we get

$\rm \: = \: - \dfrac{ {x}^{2} }{2} + 2 \times \dfrac{1}{2}log\bigg |\dfrac{x - 1}{x + 1} \bigg| + c$

$\rm \: = \: - \dfrac{ {x}^{2} }{2} + log\bigg |\dfrac{x - 1}{x + 1} \bigg| + c$

Hence,

$\boxed{ \rm{ \:\rm \: \displaystyle\int\rm \frac{ {x}^{3} - x - 2}{1 - {x}^{2} }dx= \: - \dfrac{ {x}^{2} }{2} + log\bigg |\dfrac{x - 1}{x + 1} \bigg| + c \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ {x}^{2} + {a}^{2} } = \dfrac{1}{a} {tan}^{ - 1} \dfrac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log |x + \sqrt{ {x}^{2} + {a}^{2}} | + c}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

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