Question

determine the indefinite integral. use capital c for the free constant.
$$\int(3 x^{4}-5 x^{2}+2) d x=$$

Explanation:

Step1: Apply integral sum rule

$$\int(3x^4 - 5x^2 + 2)dx=\int3x^4dx-\int5x^2dx+\int2dx$$

Step2: Use constant multiple rule

$$=3\int x^4dx - 5\int x^2dx+2\int dx$$

Step3: Apply power rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n

eq - 1)$ and $\int dx=x + C$
For $\int x^4dx$, $n = 4$, so $\int x^4dx=\frac{x^{4+1}}{4 + 1}=\frac{x^5}{5}$; for $\int x^2dx$, $n=2$, so $\int x^2dx=\frac{x^{2 + 1}}{2+1}=\frac{x^3}{3}$; and $\int dx=x$.
$$=3\times\frac{x^5}{5}-5\times\frac{x^3}{3}+2x+C$$

Step4: Simplify the expression

$$=\frac{3}{5}x^5-\frac{5}{3}x^3 + 2x+C$$

Answer:

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