Question

Determine the following integrals

Answer 1

$\large\underline{\sf{Given \:Question - }}$

$\sf \: Evaluate \: \displaystyle\int\rm {3}^{x}. {x}^{2} \: dx$

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

Given integral is

$\rm :\longmapsto\: \: \displaystyle\int\rm {3}^{x}. {x}^{2} \: dx$

We know, Integration by Parts,

$\boxed{ \sf \: \displaystyle\int\rm uv \: x = u\displaystyle\int\rm vdx - \displaystyle\int\rm \bigg(\dfrac{d}{dx}u\displaystyle\int\rm vdx \bigg) dx}$

So,

Here,

$\boxed{ \sf \: u \: = \: {x}^{2}} \: \: and \: \: \boxed{ \sf \: v \: = \: {3}^{x}}$

So, using this result, we get

$\rm \: = \: {x}^{2} \displaystyle\int\rm {3}^{x} dx - \displaystyle\int\rm \bigg(\dfrac{d}{dx} {x}^{2} \displaystyle\int\rm {3}^{x} dx \bigg) dx$

$\rm \: = \: {x}^{2}\dfrac{ {3}^{x} }{log3} - \displaystyle\int\rm 2x \times \dfrac{ {3}^{x} }{log3} \: dx$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2}{log3}\displaystyle\int\rm x .{3}^{x} \: dx$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2}{log3}\bigg[x\displaystyle\int\rm {3}^{x} \: dx - \displaystyle\int\rm \bigg(\dfrac{d}{dx}x\displaystyle\int\rm {3}^{x} dx \bigg) dx\bigg]$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2}{log3}\bigg[x\dfrac{ {3}^{x} }{log3} - \displaystyle\int\rm \bigg(\dfrac{ {3}^{x} }{log3}\bigg) dx\bigg]$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2}{log3}\bigg[x\dfrac{ {3}^{x} }{log3} - \dfrac{1}{log3}\displaystyle\int\rm \bigg( {3}^{x} \bigg) dx\bigg]$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2}{log3}\bigg[x\dfrac{ {3}^{x} }{log3} - \dfrac{1}{log3} \times \dfrac{ {3}^{x} }{log3}\bigg] + c$

$\rm \: = \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2x. {3}^{x} }{ {(log3)}^{2} } + \dfrac{2. {3}^{x} }{ {(log3)}^{3} } + c$

Hence,

$\boxed{ \sf \: \displaystyle\int\rm {x}^{2} . {3}^{x}dx \: \: = \bf \: \dfrac{ {x}^{2} {3}^{x} }{log3} - \dfrac{2x. {3}^{x} }{ {(log3)}^{2} } + \dfrac{2. {3}^{x} }{ {(log3)}^{3} } + c}$

Additional Information :-

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