Question

Evaluate the following integrals...
.
.
.
.​

Answer 1

Step-by-step explanation:

We have,

$\int \bigg \{ ( \tan(x) + \cot(x))^{2} + \frac{1}{x} + {e}^{x} \bigg\}dx$

$= \int \bigg \{ \tan^{2} (x) + \cot^{2} (x) + 2 + \frac{1}{x} + {e}^{x} \bigg\}dx$

$= \int \bigg \{ \tan^{2} (x) + 1+ \cot^{2} (x) + 1 + \frac{1}{x} + {e}^{x} \bigg\}dx$

$= \int \bigg \{ \sec^{2} (x) + \cosec^{2} (x) + \frac{1}{x} + {e}^{x} \bigg\}dx$

$= \int \sec^{2} (x)dx + \int \cosec^{2} (x) dx+ \int \frac{1}{x} \: dx + \int{e}^{x} dx$

$= \tan (x) - \cot (x)+ \ln(x) + {e}^{x} + c$

Answer 2

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

Evaluate the following integral :-

$\displaystyle\int\sf \bigg[{(tanx + cotx)}^{2} + \frac{1}{x} + {e}^{x}\bigg] \: dx$

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

Given integral is

$\rm :\longmapsto\:\displaystyle\int\sf \bigg[{(tanx + cotx)}^{2} + \frac{1}{x} + {e}^{x}\bigg] \: dx$

We know,

$\boxed{ \bf{ \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \: }}$

So, using this, we get

$\rm \:=\displaystyle\int\sf \bigg[ {tan}^{2}x + {cot}^{2}x + 2tanxcotx + \frac{1}{x} + {e}^{x} \bigg]dx$

We know,

$\boxed{ \bf{ \:cotx = \frac{1}{tanx}}}$

So, using this, we get

$\rm \:=\displaystyle\int\sf \bigg[ {tan}^{2}x + {cot}^{2}x + 2tanx \times \frac{1}{tanx} + \frac{1}{x} + {e}^{x} \bigg]dx$

$\rm \:=\displaystyle\int\sf \bigg[ {tan}^{2}x + {cot}^{2}x + 2+ \frac{1}{x} + {e}^{x} \bigg]dx$

$\rm \:=\displaystyle\int\sf \bigg[( {sec}^{2}x - 1) + ({cosec}^{2}x - 1)+ 2+ \frac{1}{x} + {e}^{x} \bigg]dx$

$\rm \:=\displaystyle\int\sf \bigg[{sec}^{2}x - 1 +{cosec}^{2}x -1+ 2+ \frac{1}{x} + {e}^{x} \bigg]dx$

$\rm \:=\displaystyle\int\sf \bigg[{sec}^{2}x +{cosec}^{2}x+ \frac{1}{x} + {e}^{x} \bigg]dx$

$\rm \:=\displaystyle\int\sf {sec}^{2}x dx+\displaystyle\int\sf {cosec}^{2}xdx+ \displaystyle\int\sf \frac{1}{x}dx +\displaystyle\int\sf {e}^{x}dx$

$\rm \:  =  \: tanx - cotx + log(x) + {e}^{x} + c$

Additional Information :-

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