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I'm having trouble computing the integral: $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective.

Any suggestions are welcome. Thanks.

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    http://www.wolfr$a$malpha.com/input/?i=int+%7B+sin+x+%2F%7Bcosx+%2B+sinx%7D. wolframaplha.com it is a helpful site for computations2012-08-09

8 Answers 8

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Let $I:=\int\frac{\cos x}{\cos x+\sin x}dx$ and $J:=\int\frac{\sin x}{\cos x+\sin x}dx$. Then $I+J=x + C$, and $I-J=\int\frac{\cos x-\sin x}{\cos x+\sin x}dx=\int\frac{u'(x)}{u(x)}dx,$ where $u(x)=\cos x+\sin x$. Now we can conclude.

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    This was essentially the important thing to notice for A putnam question asking to evaluate $\int_{0}^{\pi/2} \frac{1}{1+\tan^{\sqrt{2}}(x)}dx$ in which case $I=J$2016-08-30
41

Hint: $\sqrt{2}\sin(x+\pi/4)=\sin x +\cos x$, then substitute $x+\pi/4=z$

32

You can do this without thinking: use the Weierstrass substitution to reduce the integral to a rational function, and integrate that as usual.

25

We can write the integrand as $\begin{equation*} \frac{1}{1+\cot x} \end{equation*}$ and use the substitution $u=\cot x$. Since $du=-\left( 1+u^{2}\right) dx$ we reduce it to a rational function

$\begin{equation*} I:=\int \frac{\sin x}{\sin x+\cos x}dx=-\int \frac{1}{\left( 1+u\right) \left( u^{2}+1\right) }\,du. \end{equation*}$

By expanding into partial fractions and using the identities

$\begin{eqnarray*} \cot ^{2}x+1 &=&\csc ^{2}x \\ \arctan \left( \cot x\right) &=&\frac{\pi }{2}-x \\ \frac{\csc x}{1+\cot x} &=&\frac{1}{\sin x+\cos x} \end{eqnarray*}$

we get

$\begin{eqnarray*} I &=&-\frac{1}{2}\int \frac{1}{1+u}-\frac{u-1}{u^{2}+1}\,du \\ &=&-\frac{1}{2}\ln \left\vert 1+u\right\vert +\frac{1}{4}\ln \left( u^{2}+1\right) -\frac{1}{2}\arctan u +C\\ &=&-\frac{1}{2}\ln \left\vert 1+\cot x\right\vert +\frac{1}{4}\ln \left( \cot ^{2}x+1\right) -\frac{1}{2}\arctan \left( \cot x\right) +C \\ &=&-\frac{1}{2}\ln \left\vert 1+\cot x\right\vert +\frac{1}{4}\ln \left( \csc ^{2}x\right) +\frac{1}{2}x+\text{ Constant} \\ &=&\frac{1}{2}x-\frac{1}{2}\ln \left\vert \sin x+\cos x\right\vert +\text{ Constant.} \end{eqnarray*}$

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Write the numerator (here $\sin x$) as a linear combination of the denominator and the derivative of the denominator: $A(\sin x+ \cos x) + B( \cos x- \sin x) = \sin x$ Solve for $A$ and $B$ and split the fraction accordingly. Integrating give a linear term and an $\ln$ This method generally works for $\frac{Asinx+Bcosx}{Csinx+Dcosx}$ where $A$ and $B$ not both zero (one of them can be zero as in this post), and $C$ and $D$ certainly not both zero at the same time.

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$= \frac{1}{2} \cdot \int \frac{\sin(x) + \sin(x)}{\sin(x) + \cos(x)} dx = \frac{1}{2} \int \frac{\sin(x) + \cos(x) + \sin(x) - \cos(x) }{\sin(x) + \cos(x)} dx $ $= \frac{x}{2} - \frac{1}{2} \int \frac{\cos(x) - \sin(x)}{\sin(x) + \cos(x)} dx$ Let $u = \sin(x) + \cos(x)$ $ \implies \frac{x}{2} - \frac{1}{2} \cdot \log \left| \sin(x) + \cos(x) \right| + C$

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    @amad27 No worry. Happens often on MSE. It's happened to me before. Be well my friend. ;-) $A$nd +1 an$y$wa$y$2015-08-11
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Just for some new ideas! I would reccomend a completely different method. This method uses the Gudermannian $\text{gd}$ function. So you would substitute $x=\text{gd}(a);\text{d}x=\text{sech}\space a\text{d}a$ That transforms the integral into:

$\int \frac{\tanh a}{\tanh a+\text{sech}\space a}(\text{sech}\space a)\mathrm da$

Through some hyperbolic trig properties, we get (correct me if I'm wrong)

$\int {\frac{1}{\cosh a+\coth a}}\text{d}a$

You could probably take it from here

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    I am giving you a bounty because the idea is really impressive:i.e. new2015-12-01
2

So we have the integral

$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$

A little bit more general solution would be to substitute $\sin(x) + \cos(x) = k\cos(\phi+x)$:

$$\int \frac{1}{k}\frac{\sin(x)}{\cos(\phi+x)}\mathrm dx.$$

We know this is true from elementary trigonometry. Now if we know the logarithmic derivative : $\frac{\partial \ln(g(x))} {\partial x} = \frac{g'(x)}{g(x)}$ we are almost done immediately without even needing to do any substitution, however of course we need to determine $\phi$ and add a constant of integration, but that is a rather easy exercise.