warm up 4/10/2025 (notes 4.1)\nintegrate: ∫(2x + 4)dx\n○ f(x)=2\n○ f(x)=x² + 4x + c\n○ f(x)=x² + c\n○…

Explanation:

Step1: Apply sum - rule of integration

$\int(2x + 4)dx=\int 2x dx+\int 4dx$

Step2: Integrate $\int 2x dx$

Using the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $y = 2x$, $n = 1$. So $\int 2x dx=2\times\frac{x^{1+1}}{1 + 1}=x^{2}$

Step3: Integrate $\int 4dx$

Since $\int kdx=kx + C$ (where $k$ is a constant), for $k = 4$, $\int 4dx=4x$

Step4: Combine the results

$\int(2x + 4)dx=x^{2}+4x + C$

Answer:

$f(x)=x^{2}+4x + c$