Ответ:
[tex]\boxed{ \boldsymbol{ \displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \dfrac{2}{3} \sin \sqrt{3x + 1} + C} }[/tex]
Примечание:
По таблице интегралов:
[tex]\boxed{\displaystyle \int \cos x \ dx = \sin x + C}[/tex]
Пошаговое объяснение:
[tex]\displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \dfrac{2}{3} \sin \sqrt{3x + 1} + C[/tex]
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Замена: [tex]t = \sqrt{3x + 1}[/tex]
[tex]dt = d(\sqrt{3x + 1}) \ dx[/tex]
[tex]dt = \dfrac{(3x + 1)' \ dx}{2\sqrt{3x + 1} }[/tex]
[tex]dt = \dfrac{3 \ dx}{2\sqrt{3x + 1} }[/tex]
[tex]dt = \dfrac{3 \ dx}{2t} \Longrightarrow \boxed{ dx = \frac{2}{3}\cdot t \ dt}[/tex]
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[tex]\displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \int {\dfrac{2t\cos t }{ 3t} } \, dt = \frac{2}{3} \int \cos t \ dt = \dfrac{2}{3} \sin t + C =[/tex]
[tex]= \dfrac{2}{3} \sin \sqrt{3x + 1} + C[/tex]
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Answers & Comments
Ответ:
[tex]\boxed{ \boldsymbol{ \displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \dfrac{2}{3} \sin \sqrt{3x + 1} + C} }[/tex]
Примечание:
По таблице интегралов:
[tex]\boxed{\displaystyle \int \cos x \ dx = \sin x + C}[/tex]
Пошаговое объяснение:
[tex]\displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \dfrac{2}{3} \sin \sqrt{3x + 1} + C[/tex]
--------------------------------------------
Замена: [tex]t = \sqrt{3x + 1}[/tex]
[tex]dt = d(\sqrt{3x + 1}) \ dx[/tex]
[tex]dt = \dfrac{(3x + 1)' \ dx}{2\sqrt{3x + 1} }[/tex]
[tex]dt = \dfrac{3 \ dx}{2\sqrt{3x + 1} }[/tex]
[tex]dt = \dfrac{3 \ dx}{2t} \Longrightarrow \boxed{ dx = \frac{2}{3}\cdot t \ dt}[/tex]
-----------------------------------------------
[tex]\displaystyle \int {\dfrac{\cos\sqrt{3x + 1} }{\sqrt{3x + 1}} } \, dx = \int {\dfrac{2t\cos t }{ 3t} } \, dt = \frac{2}{3} \int \cos t \ dt = \dfrac{2}{3} \sin t + C =[/tex]
[tex]= \dfrac{2}{3} \sin \sqrt{3x + 1} + C[/tex]