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$ \int \frac{dx}{1-\tan(x)} $ Please help me to solve this problem as I'm trying this since last 1 day...

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    Duplicate of [Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)} \mathrm{d} x$](http://math.stackexchange.com/questions/180744/compute-int-frac-sinx-sinx-cosx-mathrm-dx)2012-08-29

3 Answers 3

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$\int \frac 1{1-\tan(x)}dx=\int \frac {\cos(x)}{\cos(x)-\sin(x)}dx$ $=\frac 12\int \frac {\cos(x)+\sin(x)+\cos(x)-\sin(x)}{\cos(x)-\sin(x)}dx$ $=\frac 12\int 1+\frac {\cos(x)+\sin(x)}{\cos(x)-\sin(x)}dx$ With the result : $\int \frac 1{1-\tan(x)}dx=C+\frac {x-\log(\cos(x)-\sin(x))}2$

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Hint: Since your integral is as $\int\frac{\cos(x)\,dx}{\cos(x)-\sin(x)}=\int R(\sin(x),\cos(x))\,dx$ wherein $R$ is rational function respect to $\sin(x)$ and $\cos(x)$; you can use the following change of variable: $t=\tan(x/2), -\pi

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    Nice hi$n$t! I hope you read this upon awakening fro$m$ a peaceful slu$m$ber! ;-)2013-03-14
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An alternative is to use the substitution $t=\tan x$ and then expand into partial fractions. $\begin{equation*} dt=\left( 1+\tan ^{2}x\right) dx=\left( 1+t^{2}\right) dx \end{equation*}$

$\begin{equation*} I=\int \frac{1}{1-\tan x}dx=\int \frac{1}{\left( 1-t\right) \left( 1+t^{2}\right) }\,dt. \end{equation*}$ Since $\begin{equation*} \frac{1}{\left( 1-t\right) \left( 1+t^{2}\right) }=\frac{1}{2\left( 1-t\right) }+\frac{1}{2}\frac{t}{1+t^{2}}+\frac{1}{2}\frac{1}{1+t^{2}}, \end{equation*}$ we have $\begin{eqnarray*} I &=&\frac{1}{2}\int \frac{1}{1-t}\,dt+\frac{1}{2}\int \frac{t}{1+t^{2}}\,dt+ \frac{1}{2}\int \frac{1}{1+t^{2}}\,dt \\ &=&-\frac{1}{2}\ln \left\vert 1-t\right\vert +\frac{1}{4}\ln \left\vert 1+t^{2}\right\vert +\frac{1}{2}\arctan t+C \\ &=&-\frac{1}{2}\ln \left\vert 1-\tan x\right\vert +\frac{1}{4}\ln \left\vert 1+\tan ^{2}x\right\vert +\frac{x}{2}+C \\ &=&-\frac{1}{2}\ln \left\vert 1-\tan x\right\vert +\frac{1}{2}\ln \left\vert \sec x\right\vert +\frac{x}{2}+C. \end{eqnarray*}$ This can be written as $\begin{eqnarray*} I &=&-\frac{1}{2}\ln \left\vert \frac{1-\tan x}{\sec x}\right\vert +\frac{x}{ 2}+C \\ &=&-\frac{1}{2}\ln \left\vert \cos x-\sin x\right\vert +\frac{x}{2}+C. \end{eqnarray*}$