What is wrong with this integration of $ \int_0^{2\pi}\sin x /(1 + A \sin x)$

Question

A worked solution of the integral has been provided as an answer to a previous question.

But I am still unclear why I get the wrong answer from the following method which uses a formula for the definite integral provided by Wolfram Alpha.

Wolfram Alpha indicates the following solution formula:-

$$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = (1/A)\left( x - \frac{2 \tan^{-1} \left( \frac{A + \tan{(x/2)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + constant\right)^{2\pi}_0 $$

My thinking is that substituting for $x$ by $2 \pi$ and by $0$ leads to the corresponding values of $\tan(x/2)$ being the same, namely $\tan(\pi) = \tan(0) = 0$. It follows therefore that the fractional term:- $$ \frac{2 \tan^{-1} \left( \frac{A + \tan{(x/2)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} $$ has the same value ( let us call it $Q$ ) for $x=2\pi$ and $x=0.$ Therefore the definite integral becomes

$$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = (1/A)\left( (2\pi - Q + constant) - (0 - Q + constant) \right) = 2\pi/A. $$

For $A = 0.2$ we derive the result $$ \int_0^{2\pi} \frac{\sin x}{1 + 0.2 \sin x} dx = 10 \pi. $$

However Wolfram Alpha gives the answer as $-0.64782$ (approx.) which is reasonable looking at the graph and agrees with the formula provided in @abel's answer to my previous question.

So what is wrong with my reasoning? I realize that the $\tan$ function is not continuous over the range $0,2\pi$ but I don't know what that implies for the analysis of the definite integral. What rules or tricks should I apply in order to use the Wolfram Alpha formula? Must I always beware the appearance of $\tan()$ functions in integrand formulae?

Answer 1

The main problem is that the Fundamental Theorem of Calculus only applies if the antiderivative you apply it to is continuous. If you graph your antiderivative, you will notice that (whatever the value of $a$), it is discontinuous at $x=\pi$.

To make $$ \arctan{\left( \frac{A+\tan{\frac{1}{2}x}}{\sqrt{1-A^2}} \right)} $$ continuous, you have to add something like $\pi\lfloor x/(2\pi)+1/2 \rfloor $, which cancels out the discontinuity in the arctangent (as $\tan{\frac{1}{2}x}$ tends to $+\infty$ as $x \uparrow \pi$, but $-\infty$ as $x \downarrow \pi$, so the arctangent has a discontinuity as it switches between $-\pi/2$ and $\pi/2$). You will find that when you evaluate the integral using the antiderivative, you do get the correct answer.

Answer 2

Using the guidance provided by Chappers and Hans I have come up with this approach.

The Wolfram Alpha solution is:-

$$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = (1/A)\left( x - \frac{2 \tan^{-1} \left( \frac{A + \tan{(x/2)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + C\right)^{2\pi}_0 $$

The antiderivative (on the RHS of the equation above) cannot be evaluated by plugging in values for the limiting values of $x=0$ and $x=2\pi$ because the $\tan^{-1}$ term in the anti-derivative function is not continuous over the range $0..2\pi$

See this plot.

However the $\tan^{-1}$ term is cyclic with a period of $\pi$. Therefore the definite integral over $0..2\pi$ is the same as the definite integral over any other range of length $2\pi$ or two times the definite integral over any range of length $\pi$. We proceed by selecting one particular range: $-\pi/2..\pi/2$ over which the $\tan^{-1}$ term is continuous. We will need to multiply the expression by two to compensate for the difference in ranges.

Proceeding thus... $$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = Q = (2*1/A)\left( x - \frac{2 \tan^{-1} \left( \frac{A + \tan{(x/2)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + C\right)_{-\pi/2}^{+\pi/2} $$

$$Q = (2/A) \left[ \left( \pi/2- \frac{2 \tan^{-1} \left( \frac{A + \tan{(+\pi/4)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + C\right) - \left( (-\pi/2) - \frac{2 \tan^{-1} \left( \frac{A + \tan{(-\pi/4)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + C\right) \right] $$

$$Q = (2/A) \left[ \pi - \frac{ 2\tan^{-1} \left( \frac{A + 1}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + \frac{ 2\tan^{-1} \left( \frac{A -1}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} \right] $$

$$Q = \frac{2}{A} \left[ \pi + \frac{ 2\tan^{-1} \left( \frac{A - 1}{\sqrt{(1-A^2)}}\right) -2\tan^{-1} \left( \frac{A + 1}{\sqrt{(1-A^2)}}\right) } {\sqrt{(1-A^2)}} \right] $$

It happens that (for $0<A<1$) (tested in Excel, see Note at bottom of answer) $$ \tan^{-1} \left( \frac{A - 1}{\sqrt{(1-A^2)}}\right) -\tan^{-1} \left( \frac{A + 1}{\sqrt{(1-A^2)}}\right) =\tan^{-1} (-\infty) = - \pi/2 $$

and so we get

$$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = Q = \frac{2\pi}{A} \left[ 1- \frac{1}{\sqrt{(1-A^2)}} \right] $$

which is the same as the answer derived by @abel in answer to the previous question and for which, with $A=0.2$ we get the result $Q=-0.64782$ (approx.).

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Note No proof is provided here for the assertion that ( for $0<A<1$ ) $$ \tan^{-1} \left( \frac{A - 1}{\sqrt{(1-A^2)}}\right) -\tan^{-1} \left( \frac{A + 1}{\sqrt{(1-A^2)}}\right) =\tan^{-1} (-\infty) = - \pi/2 $$ However it can be checked (a) by testing in a spreadsheet such as Excel or (b) using the inverse-tangent addition theorem:- $$ \tan^{-1} x \pm \tan^{-1} y = \tan^{-1} \left( \frac{x \pm y}{1 \mp xy} \right). $$