What is the primitive of $\dfrac{1}{\sin x+1}$?

Question

I hope you are well I try to find the infinite primitive of $\dfrac{1}{\sin x+1}$ without using the méthode of substituting by $\tan \frac x2$.

So I multiply and divide by $\sin x - 1$ and simplify until having The result: $\tan x -\dfrac{1}{\cos x}$ But the professor said that my méthode is not totally correct because I can't divide by $1-\sin x$ because this expression can be null , so is there any methods except of the substitution using $\tan \frac x2$ to find this primitive?

Thank you for reading the question and for helping.

@HarishChandraRajpoot - It would be more helpful to give a link. — mr_e_man, Apr 27 '23 at 23:16
@wissal - Indeed, the integrand is defined for all $x\neq-\pi/2$ $(\bmod2\pi)$, but your result is not defined at $x=+\pi/2$, so it's not quite correct. However, setting aside these singular points, your result is equivalent to $-\cos(x)/(1+\sin(x))$, so I suggest finding a way to derive that directly. — mr_e_man, Apr 27 '23 at 23:31
Actually any solution involving $\tan(x/2)$ has the same problem, since that's undefined at $x=\pi$. — mr_e_man, May 03 '23 at 16:08

score 8 · Answer 1 · answered Apr 27 '23 at 23:32

8

Hint: $$\frac{1}{\sin x+1}=\frac1{\cos(\frac\pi2-x)+1}=\frac1{2\cos^2(\frac\pi4-\frac x2)}= \frac12\sec^2(\frac\pi4-\frac x2) $$

answered Apr 27 '23 at 23:32

Quanto

97,352

I don't know what the OP had in mind with $\tan(x/2)$, but your answer gives $\tan(x/2-\pi/4)$ so it's similar. – mr_e_man May 03 '23 at 16:12

score 1 · Answer 2 · answered Apr 28 '23 at 03:53

1

$$ \begin{aligned} \int \frac{1}{1+\sin x} d x & =\int \frac{1-\sin x}{\cos ^2 x} d x \\ & =\int \sec ^2 x d x-\int \sec x \tan x d x \\ & =\tan x-\sec x+C \end{aligned} $$

answered Apr 28 '23 at 03:53

Lai

20,421

But how to find (cos(x))^2? – Alia Apr 28 '23 at 08:29
$1-\sin^2x =\cos^2 x$, got it? – Lai Apr 28 '23 at 08:46
1

But you can't divide by 1-(sinx)^2 because it can equals to 0 – Alia Apr 28 '23 at 23:48
This is exactly the "not totally correct" method suggested in the OP. – mr_e_man May 03 '23 at 15:44

What is the primitive of $\dfrac{1}{\sin x+1}$?

2 Answers2