warm up 4/10/2025 (notes 4.1)\nintegrate: ∫(x⁴ + 3x² - x + 4) dx\no x⁵/5 + x³ - x²/2 + 4x + c\no x⁵/5 + x³…

Explanation:

Step1: Apply power - rule for integration

The power - rule for integration is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int x^{4}dx$, using the power - rule with $n = 4$, we get $\frac{x^{4+1}}{4 + 1}=\frac{x^{5}}{5}$. For $\int3x^{2}dx$, since $\int kf(x)dx=k\int f(x)dx$ ($k$ is a constant), and $\int x^{2}dx=\frac{x^{2 + 1}}{2+1}=\frac{x^{3}}{3}$, then $\int3x^{2}dx=3\times\frac{x^{3}}{3}=x^{3}$. For $\int(-x)dx=-\int xdx$, and using the power - rule with $n = 1$, we have $-\frac{x^{1+1}}{1+1}=-\frac{x^{2}}{2}$. For $\int4dx$, since $\int kdx=kx + C$ ($k$ is a constant), then $\int4dx=4x$.

Step2: Combine the results

$\int(x^{4}+3x^{2}-x + 4)dx=\int x^{4}dx+\int3x^{2}dx-\int xdx+\int4dx=\frac{x^{5}}{5}+x^{3}-\frac{x^{2}}{2}+4x + C$.

Answer:

$\frac{x^{5}}{5}+x^{3}-\frac{x^{2}}{2}+4x + c$ (the first option)