![{\displaystyle \int _{0}^{\varpi }{\frac {\operatorname {Sin} .(p+q)x-\operatorname {Sin} .(p-q)x}{2\operatorname {Sin} .x}}\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e873af9605e3489325452eb097465d65b62ffbe9)
ou
![{\displaystyle \int _{0}^{\varpi }{\frac {\operatorname {Cos} .px.\operatorname {Sin} .qx}{\operatorname {Sin} .x}}\operatorname {d} x=\varpi .\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6069af96d43d33a400c82f2f2785782eb99997)
(4)
De là on peut conclure, en particulier, 1.o que, quel que soit le nombre entier positif
les intégrales, en nombre infini
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,\ \int {\frac {\operatorname {Cos} .(r+1)x.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b79d211873f0f8df8ad135065b42fa5f20c39a)
![{\displaystyle \int {\frac {\operatorname {Cos} .(r+2)x.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef05215ca606a8a1e9e6c45b35608b3f7dd8022)
ainsi que les intégrales, en nombre infini,
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+2)x}{\operatorname {Sin} .x}}\operatorname {d} x,\ \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+4)x}{\operatorname {Sin} .x}}\operatorname {d} x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13bfaa18e3c818c5e8f990441f97e122910af917)
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+6)x}{\operatorname {Sin} .x}}\operatorname {d} x,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2fa83118a8012520fe4fafd8093c54abce4eb)
prises entre les limites
et
seront nulles ; et qu’il en sera encore de même des
intégrales
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .x}{\operatorname {Sin} .x}}\operatorname {d} x,\ \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .2x}{\operatorname {Sin} .x}}\operatorname {d} x,\ldots \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r-1)x}{\operatorname {Sin} .x}}\operatorname {d} x\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fcd5f954437b8372cc2b697bb156d34968a451)
tandis que les intégrales, en nombre infini,
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+1)x}{\operatorname {Sin} .x}}\operatorname {d} x,\ \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+3)x}{\operatorname {Sin} .x}}\operatorname {d} x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3602f2460e830c3fcbe057a25d36e85defaf13d)
![{\displaystyle \int {\frac {\operatorname {Cos} .rx.\operatorname {Sin} .(r+5)x}{\operatorname {Sin} .x}}\operatorname {d} x,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ca2dc4f4824bc0344ee8dceb9627f3a967e747)
prises entre les mêmes limites, seront toutes égales à ![{\displaystyle \varpi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/708f6611303e54eefbedd289cd48ca2ed16af127)
2.o Que, si le nombre entier positif
est pair, les
intégrales
![{\displaystyle \int {\frac {\operatorname {Cos} .0x.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,\ \int {\frac {\operatorname {Cos} .2x.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,\ldots \int {\frac {\operatorname {Cos} .(r-2)x.\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\operatorname {d} x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe31f3f2ff7c9a9f6cb4c39e3ef9db293d5c222)
prises toujours entre les limites
et
seront nulles, tandis que les
autres intégrales