example 2 find each indefinite integral. a. ∫(x + 2)dx b. ∫(3x⁴ - 5x² + x)dx

Explanation:

Step1: Recall integral rules

The integral of a sum is the sum of integrals, i.e., $\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$, and $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$).

Step2: Solve part a

For $\int(x + 2)dx=\int xdx+\int 2dx$. Since $\int xdx=\frac{x^{1+1}}{1 + 1}=\frac{x^{2}}{2}$ and $\int 2dx=2x$, then $\int(x + 2)dx=\frac{x^{2}}{2}+2x+C$.

Step3: Solve part b

For $\int(3x^{4}-5x^{2}+x)dx=3\int x^{4}dx-5\int x^{2}dx+\int xdx$. We know that $\int x^{4}dx=\frac{x^{4 + 1}}{4+1}=\frac{x^{5}}{5}$, $\int x^{2}dx=\frac{x^{2+1}}{2 + 1}=\frac{x^{3}}{3}$, $\int xdx=\frac{x^{1+1}}{1+1}=\frac{x^{2}}{2}$. So $3\int x^{4}dx-5\int x^{2}dx+\int xdx=3\times\frac{x^{5}}{5}-5\times\frac{x^{3}}{3}+\frac{x^{2}}{2}+C=\frac{3x^{5}}{5}-\frac{5x^{3}}{3}+\frac{x^{2}}{2}+C$.

Answer:

a. $\frac{x^{2}}{2}+2x+C$ b. $\frac{3x^{5}}{5}-\frac{5x^{3}}{3}+\frac{x^{2}}{2}+C$