Question

Solve the following

$\int \: \frac{dx}{1 + tanx}$
​

Answer 1

answer

your answer this photo

Answer 2

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

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 :-

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