Question
Primitive de \( \frac{1}{\sin x} \)
Ask by Estrada Page. in Togo
Feb 20,2025
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Answer
Une primitive de \( \frac{1}{\sin x} \) est \( \ln |\tan(x/2)| + C \).
Solution
Nous cherchons une primitive de 1/sin x, c'est-à-dire de la fonction csc x. Une primitive connue de csc x est :
ln |tan(x/2)| + C
Pour vérifier cette formule, nous pouvons dériver ln |tan(x/2)| :
1. Soit F(x) = ln |tan(x/2)|. Sa dérivée est donnée par la règle de dérivation d'un logarithme :
F′(x) = (1/ tan(x/2)) · (tan(x/2))′.
2. Le dérivé de tan(x/2) est :
(tan(x/2))′ = (1/2) · sec²(x/2).
3. On obtient alors :
F′(x) = (1/ tan(x/2)) · (1/2) · sec²(x/2)
= (1/2) · (sec²(x/2)/tan(x/2)).
4. Exprimons maintenant tan et sec en fonctions de sin et cos :
tan(x/2) = sin(x/2) / cos(x/2) et sec²(x/2) = 1/ cos²(x/2).
Ainsi :
sec²(x/2)/ tan(x/2) = (1/ cos²(x/2)) / (sin(x/2)/cos(x/2)) = 1/( sin(x/2) cos(x/2) ).
5. Nous avons alors :
F′(x) = (1/2) · [1/( sin(x/2) cos(x/2) )].
6. Notons que l'identité trigonométrique 2 sin(x/2) cos(x/2) = sin x permet d'écrire :
1/(2 sin(x/2) cos(x/2)) = 1/sin x.
Ainsi, F′(x) = 1/sin x.
Par conséquent, ln |tan(x/2)| + C est bien une primitive de 1/sin x.
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Simplify this solution Mind Expander
Did you know that the integral of \( \frac{1}{\sin x} \) leads us to a commonly known function called the cosecant? This function's integral can be expressed as \( \ln | \csc x - \cot x | + C \). It's fascinating how integrals link back to functions that appear throughout trigonometry and other areas of mathematics! If you're tackling this integral, be mindful of the domain restrictions! Since \( \sin x \) can be zero at multiples of \( \pi \), the function \( \frac{1}{\sin x} \) becomes undefined at those points. Always remember to check for discontinuities when integrating to avoid those pesky pitfalls that might sneak into your calculations!
