# Integration by substitution fail for $\int \frac{1}{(1+\sin x)}\, \mathrm dx$.

We tried to do an integral with the universal trigonometric substitution $$\int \frac{1}{(1+\sin x)}\, \mathrm dx$$

Meaning, we substituted: $$t = \tan \frac{x}{2} \Rightarrow$$

$$\int \frac{1}{(1+\sin x)}\, \mathrm dx = \int \frac{\frac{2}{1+t^2}}{1+\frac{2t}{1+t^2}}\, \mathrm dt = \int \frac{2}{(1+t)^2}\, \mathrm dt = \frac{-2}{1+t} = \frac{-2}{1+\tan \frac{x}{2}} + C$$

But the answer is: $$\tan x - \frac{1}{\cos x} + C$$

What did we do wrong?

• I think your approach is correct. Manipulate a little bit your final result and you will come up with the answer. Commented Jan 21, 2014 at 10:44

\begin{align} \frac{-2}{1+\tan \frac{x}{2}} &= \frac{-2\cos \frac{x}{2}}{\cos\frac{x}{2} + \sin \frac{x}{2}}\\ &= \frac{-2\cos \frac{x}{2}(\cos \frac{x}{2}-\sin\frac{x}{2})}{(\cos \frac{x}{2}+\sin\frac{x}{2})(\cos \frac{x}{2}-\sin\frac{x}{2})}\\ &= \frac{\sin x - 2 \cos^2 \frac{x}{2}}{\cos x}\\ &= \tan x - \frac{\cos x + 1}{\cos x}\\ &= \tan x - \frac{1}{\cos x} - 1 \end{align}

You did nothing wrong, you just got another representation of the same family of functions.

• when I put the functions in WolframAlpha, it say's that the functions are not equal for every x... Commented Jan 21, 2014 at 10:44
• You have different constants. Plug in $x = 0$ to see that the main part of your primitive and the answer's is $1$ there. Differentiate to see that both are primitives of the integrand, so the difference is constant. Commented Jan 21, 2014 at 10:47
• @Tomer, I'm no math whiz, so I can't directly comment on the correctness of Daniel's answer. However, Wolfram Alpha, when not given any constraints, takes the broadest possible approach to the problem. Wolfram will consider possibilities such as imaginary values for x, even if you think "x is a real number" is implied. This sort of thing is, in my experience, the most common logical error when dealing with Wolfram technologies. Commented Jan 21, 2014 at 19:20

An alternative method of integration first involves multiplying the numerator and denominator by $1-\sin x$: this gives \begin{align*} \int \frac{1}{1+\sin x} \, dx &= \int \frac{1 - \sin x}{1 - \sin^2 x} \, dx \\ &= \int \frac{1 - \sin x}{\cos^2 x} \, dx \\ &= \int \sec^2 x - \frac{\sin x}{\cos^2 x} \, dx \\ &= \tan x - \int \frac{-du}{u^2}, \quad u = \cos x, du = -\sin x \, dx \\ &= \tan x - \frac{1}{u} + C \\ &= \tan x - \sec x + C. \end{align*} To see the equivalence of this form with the expression $$-\frac{2}{1+\tan \frac{x}{2}} + C,$$ consider their difference, with $\theta = x/2$: \begin{align*} \tan x - \sec x + \frac{2}{1 + \tan \frac{x}{2}} &= \frac{\sin 2\theta - 1}{\cos 2\theta} + \frac{2}{1 + \tan \theta} \\ &= \frac{\sin 2\theta - 1}{\cos 2\theta} + \frac{2 \cos \theta}{\sin \theta + \cos \theta} \\ &= \frac{\sin 2\theta - 1 + 2 \cos\theta(\cos \theta - \sin \theta)}{\cos 2\theta} \\ &= \frac{2 \cos^2 \theta - 1}{\cos 2\theta} \\ &= 1. \end{align*} Thus their difference is constant, and both are antiderivatives.

• $$\frac{\sin x}{\cos^2 x}=\sec x \tan x$$ which integrates to $\sec x$. This is a standard result. Commented Dec 14, 2020 at 21:45

Another approach would be to first multiply the integrand by $\dfrac{1-\sin x}{1-\sin x}$ to get \begin{aligned}\int \frac{1}{1+\sin x}\,dx &= \int\frac{1-\sin x}{1-\sin^2 x}\,dx\\ &= \int\frac{1-\sin x}{\cos^2x}\,dx \\ &= \int\frac{1}{\cos^2 x}-\frac{\sin x}{\cos^2 x}\,dx \\ &= \int \sec^2x -\frac{\sin x}{\cos^2x}\,dx\\ &= \ldots\end{aligned} The first term is trivial to integrate, and the second integral requires the substitution $u=\cos x$. I leave it to you to finish things off.

Another answer is $$\int\frac{1}{1+\sin x} dx=-\frac{\cos x}{1+\sin x}+c$$ To show this we write $$\tan x-\sec x=\frac{\sin x-1}{\cos x}= \frac{(\sin x-1)\cos x}{\cos^2 x}= \frac{(\sin x-1)\cos x}{1-\sin^2 x}= \frac{(\sin x-1)\cos x}{(1-\sin x)(1+\sin x)}= -\frac{\cos x}{1+\sin x}+c$$

Let $x=\dfrac\pi2-2u$. Then $dx=-2du$ and so the integral is $$\int\frac{-2du}{1+\cos(2u)}=-\int \sec^2(u)du=-\tan u+c=\cdots$$ where $c$ is constant.
We used $\cos 2u=2\cos^2 u-1$ and $\sec x=\cos^{-1}x$.