Question

Integration of ∫(cos3x+3sinx)dx integration​

Answer 1

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

Explanation:

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

`\implies\displaystyle\sf 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\sf I = \int cos\ u (1)/(3)du + \int 3sin \ x \ dx \\\\\implies\displaystyle \sf I = \int (cos\ u)/(3) du + \int 3sin\ x \ dx \\\\\implies\displaystyle\sf I = (1)/(3)\int (cos(u))/(3) + \int 3sin(x) dx \\\\\implies\displaystyle\sf I = (1)/(3) sin(u) + C +\int 3sin(x) dx \\\\\implies\displaystyle \sf I = (1)/(3)sin(u) + C + 3\int sin(x) \ dx \\\\\implies\displaystyle\sf I = (1)/(3)sin(u) + C + 3(-cos(x)+C) \\\\\implies \underset{\blue{\sf Required\ Answer }}{\underbrace{\boxed{\boxed{\displaystyle\red{\sf I = (1)/(3)sin(3x) - 3cos(x) + C }}}}}`

Answer 2

`\displaystyle \large{(\sin 3x)/(3) - 3\cos x + C}`

Explanation:

We are given the indefinite integral:—

`\displaystyle \large{\int (\cos 3x + 3\sin x) \ dx}`

Important Formulas

`\displaystyle \large{\int f(ax+b) \ dx = (1)/(a) F(ax+b) + C}\\\displaystyle \large{\int \cos(ax) \ dx = (1)/(a) \sin (ax) + C \ \ \tt{(a \ \ is \ \ a \ \ constant.)}}\\\displaystyle \large{\int \sin x \ dx = - \cos x + C}\\\displaystyle \large{\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm \int g(x) \ dx}\\\displaystyle \large{\int kf(x) \ dx = k \int f(x) \ dx \ \ (\tt{k \ \ is \ \ a \ \ constant.})}`

Therefore, from the integral, apply the properties above:—

`\displaystyle \large{\int (\cos 3x + 3\sin x) \ dx = \int \cos 3x \ dx + \int 3 \sin x \ dx}\\\displaystyle \large{\int (\cos 3x + 3\sin x) \ dx = \int \cos 3x \ dx + 3 \int \sin x \ dx}\\\displaystyle \large{\int (\cos 3x + 3\sin x) \ dx = (1)/(3) \sin 3x + 3\cdot -\cos x + C}\\\displaystyle \large{\int (\cos 3x + 3\sin x) \ dx = (\sin 3x)/(3) - 3\cos x + C}`

Hence, the solution is:—

`\displaystyle \large \boxed{(\sin 3x)/(3) - 3\cos x + C}`