Question

Evaluate the following integrals.
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If anyone knows pls answer this fast...
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Answer 1

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

Evaluate the following integral

$\sf \: \displaystyle\int\sf {cot}^{ - 1}\bigg[\dfrac{cos6x + sin6x}{cos6x - sin6x} \bigg] \: dx$

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

The given integral is

$\rm :\longmapsto\: \sf \: \displaystyle\int\sf {cot}^{ - 1}\bigg[\dfrac{cos6x + sin6x}{cos6x - sin6x} \bigg] \: dx$

We know

$\boxed{ \bf{ \: {cot}^{1}x = {tan}^{ - 1}\bigg[\dfrac{1}{x} \bigg]}}$

So, using this we get

$\rm \:  =  \: \displaystyle\int\sf {tan}^{ - 1}\bigg[\dfrac{cos6x - sin6x}{cos6x + sin6x} \bigg] \: dx$

Divide numerator and denominator by cos6x, we get

$\rm \:  =  \: \displaystyle\int\sf {tan}^{ - 1}\bigg[\dfrac{1 - tan6x}{1 + tan6x} \bigg] \: dx$

can be rewritten as

$\rm \:  =  \: \displaystyle\int\sf {tan}^{ - 1}\bigg[\dfrac{1 - tan6x}{1 + tan6x \times 1} \bigg] \: dx$

We know,

$\boxed{ \bf{ \:{tan}^{ - 1}\bigg[\dfrac{x - y}{1 + xy} \bigg] = {tan}^{ - 1}x - {tan}^{ - 1}y}}$

So, using this identity we get

$\rm \:  =  \: \displaystyle\int\sf \bigg[{tan}^{ - 1}1 - {tan}^{ - 1}(tan6x)\bigg] \: dx$

$\rm \:  =  \: \displaystyle\int\sf \bigg[\dfrac{\pi}{4} - 6x\bigg] \: dx$

We know,

$\boxed{ \bf{ \:\displaystyle\int\sf kdx \: = \: kx + c \: \: }}$

and

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

So, using these, we get

$\rm \:  =  \: \dfrac{\pi}{4}x - 6\bigg[\dfrac{ {x}^{2} }{2} \bigg] + c$

$\rm \:  =  \: \dfrac{\pi}{4}x - {3x}^{2} + c$

Hence,

$\rm :\longmapsto\:\boxed{ \bf{ \: \sf \: \displaystyle\int\sf {cot}^{ - 1}\bigg[\dfrac{cos6x + sin6x}{cos6x - sin6x} \bigg] \: dx = \dfrac{\pi \: x}{4} - {3x}^{2} + c}}$

Additional Information :-

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