Math, asked by devansh9257, 2 months ago

Solve the following

 \int \:  \frac{dx}{1 + tanx}

Answers

Answered by anantmishra321
0

answer

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Answered by mathdude500
5

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

Given integral is

\rm \: \displaystyle\int\rm  \frac{dx}{1 + tanx}  \\

can be rewritten as

\rm \: =  \displaystyle\int\rm  \frac{dx}{1 +  \dfrac{sinx}{cosx} }  \\

\rm \: =  \displaystyle\int\rm  \frac{dx}{\dfrac{cosx + sinx}{cosx} }  \\

\rm \:  = \displaystyle\int\rm  \frac{cosx}{cosx + sinx} \: dx

can be further rewritten as

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  \frac{2cosx}{cosx + sinx} \: dx

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  \frac{cosx + cosx}{cosx + sinx} \: dx

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  \frac{cosx + cosx + sinx - sinx}{cosx + sinx} \: dx

can be re-arranged as

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  \frac{cosx + sinx - sinx + cosx}{cosx + sinx} \: dx

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  \frac{cosx + sinx }{cosx + sinx} \: dx + \dfrac{1}{2} \displaystyle\int\rm  \frac{ - sinx  + cosx}{cosx + sinx} \: dx \\

\rm \:  =  \dfrac{1}{2} \displaystyle\int\rm  1\: dx + \dfrac{1}{2} \displaystyle\int\rm  \frac{ \dfrac{d}{dx}(cosx + sinx)}{cosx + sinx} \: dx \\

We know,

\boxed{ \rm{ \:\displaystyle\int\rm k \: dx \:  =  \: kx \:  +  c \: }} \\

and

\boxed{ \rm{ \:\displaystyle\int\rm  \frac{f'(x)}{f(x)} \: dx \:  =  \: log |f(x)|  + c \:}}  \\

So, using this result, we get

\rm \:  =  \: \dfrac{1}{2}x +  \dfrac{1}{2}log |cosx + sinx|  + c \\

Hence,

\boxed{ \rm{ \:\rm \: \displaystyle\int\rm  \frac{dx}{1 + tanx}  =  \: \dfrac{1}{2}x +  \dfrac{1}{2}log |cosx + sinx|  + 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}

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