Question

∫(cos3x+3sinx)dx integration​

Answer 1

(47)83-x (cos7y) + cos34

Answer 2

`I = (1)/(3)sin(3x) - 3cos(x) + C`

Explanation:

We need to integrate the given expression. Let I be the answer .

`\implies\displaystyle I = \int (cos(3x) + 3sin(x) )dx \\\\\implies\displaystyle I = \int cos(3x) + \int sin(x)\ dx`

• Let u = 3x , then du = 3dx . Henceforth 1/3 du = dx .
• Rewrite using du and u .

`\implies\displaystyle I = \int cos\ u (1)/(3)du + \int 3sin \ x \ dx \\\\\implies\displaystyle I = \int (cos\ u)/(3) du + \int 3sin\ x \ dx \\\\\implies\displaystyle I = (1)/(3)\int (cos(u))/(3) + \int 3sin(x) dx \\\\\implies\displaystyle I = (1)/(3) sin(u) + C +\int 3sin(x) dx \\\\\implies\displaystyle I = (1)/(3)sin(u) + C + 3\int sin(x) \ dx \\\\\implies\displaystyle I = (1)/(3)sin(u) + C + 3(-cos(x)+C) \\\\\implies \boxed{\boxed{\displaystyle I = (1)/(3)sin(3x) - 3cos(x) + C }}`