Question

Evaluate the following integrals..
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Please answer fastt
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Who will give correct answer will be marked brainlist ​

Answer 1

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

Evaluate the following integral

$\displaystyle\int\sf \bigg[ {x}^{4} + {3x}^{2} - 2x - 5cosx + 7 {e}^{x} \bigg] \: dx$

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

The given integral is

$\rm :\longmapsto\:\displaystyle\int\sf \bigg[ {x}^{4} + {3x}^{2} - 2x - 5cosx + 7 {e}^{x} \bigg] \: dx$

can be rewritten as

$\rm =\displaystyle\int\sf {x}^{4}dx + 3\displaystyle\int\sf {x}^{2}dx - 2\displaystyle\int\sf xdx - 5\displaystyle\int\sf cosxdx + 7\displaystyle\int\sf {e}^{x}dx$

We know,

$\boxed{ \bf{ \:\displaystyle\int\sf {e}^{x}dx \: = \: {e}^{x} + c}}$

$\boxed{ \bf{ \:\displaystyle\int\sf cosxdx \: = \: sinx + c}}$

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

So, on using these, we get

$\rm \:  =  \: \bigg[\dfrac{ {x}^{5} }{5} \bigg] + 3\bigg[\dfrac{ {x}^{3} }{3} \bigg] - 2\bigg[\dfrac{ {x}^{2} }{2} \bigg] - 5sinx + 7{e}^{x} + c$

$\rm \:  =  \: \bigg[\dfrac{ {x}^{5} }{5} \bigg] + {x}^{3} - {x}^{2} - 5sinx + 7{e}^{x} + c$

Additional Information :-

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