Question

integrate the following​

Answer 1

Answer:

why why why why

Step-by-step explanation:

why why why

Answer 2

Given Question :-

Integrate the following :-

$\rm \: \displaystyle\int\rm \dfrac{ {a}^{6 log_{a}x } - {a}^{4 log_{a}x } }{{a}^{4 log_{a}x } - {a}^{3log_{a}x }} \: dx$

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

Given integral is

$\rm \: \displaystyle\int\rm \dfrac{ {a}^{6 log_{a}x } - {a}^{4 log_{a}x } }{{a}^{4 log_{a}x } - {a}^{3log_{a}x }} \: dx$

We know,

$\boxed{ \rm{ \:ylogx \: = \: log {x}^{y} \: }}$

So, using this result, we get

$\rm \: = \displaystyle\int\rm \dfrac{ {a}^{log_{a} {x}^{6} } - {a}^{log_{a} {x}^{4} } }{{a}^{log_{a} {x}^{4} } - {a}^{log_{a} {x}^{3} }} \: dx$

We know,

$\boxed{ \rm{ \:{a}^{log_{a} {x}^{y} } = {x}^{y} \: }}$

So, using this result, we get

$\rm \: = \: \displaystyle\int\rm \frac{ {x}^{6} - {x}^{4} }{ {x}^{4} - {x}^{3} } \: dx$

$\rm \: = \: \displaystyle\int\rm \frac{ {x}^{4}({x}^{2} - 1) }{ {x}^{3}(x - 1) } \: dx$

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

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

$\rm \: = \: \displaystyle\int\rm x(x + 1) \: dx$

$\rm \: = \: \displaystyle\int\rm ( {x}^{2} + x) \: dx$

We know,

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

So, using this result, we get

$\rm \: = \: \dfrac{ {x}^{3} }{3} + \dfrac{ {x}^{2} }{2} + c$

Hence,

$\rm\implies\boxed{ \rm{ \:\displaystyle\int\rm \dfrac{ {a}^{6 log_{a}x } - {a}^{4 log_{a}x } }{{a}^{4 log_{a}x } - {a}^{3log_{a}x }} \: dx = \: \dfrac{ {x}^{3} }{3} + \dfrac{ {x}^{2} }{2} + c \: }}$

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

Additional Information :-

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