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I've got troubles in computing the below integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$ I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ and $u=\cos(x)$ but not so effective the results were.

Any suggestions are welcomed. Thanks.

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I removed the display from your title. It's better to keep titles simple so that the main page does not break too much. –  Mariano Suárez-Alvarez Aug 9 '12 at 17:01
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wolframalpha.com/input/?i=int+%7B+sin+x+%2F%7Bcosx+%2B+sinx%7D. wolframaplha.com it is a helpful site for computations –  clark Aug 9 '12 at 17:06
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5 Answers

up vote 118 down vote accepted

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 is almost too cute for its own good :-) –  Mariano Suárez-Alvarez Aug 9 '12 at 17:06
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Nice proof. Simple and elegant. Thanks! –  Michael Li Aug 9 '12 at 17:12
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You can make this even simpler by just writing the $\sin x$ in the numerator as ${1 \over 2}(\sin x + \cos x) - {1 \over 2}(\cos x - \sin x)$. –  Zarrax Aug 9 '12 at 17:12
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@Zarrax, that magical step only helps in hiding what is going on :-) –  Mariano Suárez-Alvarez Aug 9 '12 at 18:09
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In this math education article the author describes giving the same problem to a young Terence Tao, aged 8; he gave essentially the same beautiful solution. –  Erick Wong Aug 9 '12 at 18:29
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Hint: $\sqrt{2}\sin(x+\pi/4)=\sin x +\cos x$, then substitute $x+\pi/4=z$

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Thanks. This is a new formula for me. The pity is a number of constants appears meanwhile. –  Michael Li Aug 9 '12 at 17:30
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$\sin(A+B)=\sin A \cos B+\cos A \sin B$ actually. –  Bunuelian Trick Aug 9 '12 at 17:32
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Very nice, at least if we can regard $\int \csc t \,dt$ as a "book" integral. –  André Nicolas Aug 9 '12 at 18:57
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@Michael This formula generalizes to $$A\cos x+B\sin x=\sqrt{A^{2}+B^{2}}\sin \left( x+\arctan \frac{A}{B}\right) ,$$ which is very useful in (electrical) engineering. –  Américo Tavares Aug 9 '12 at 20:05
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You can do this without thinking: use the Weierstrass substitution to reduce the integral to a rational function, and integrate that as usual.

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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 $\ln$

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