Questions: Question number 2. Find the indefinite integral ∫(2x^3+5x^2-3x+2)dx 1/2 x^4+5/3 x^3+3/2 x^2+2x+C 6x^2+C+10x-3 1/2 x^4+5/3 x^3-3x^2+2x+C 1/4 x^4+5/3 x^3-3/2 x^2+2x+C 1/2 x^4+5/3 x^3-3/2 x^2+2x+C

Solution Steps

To find the indefinite integral of the polynomial $\int (2x^3 + 5x^2 - 3x + 2) \, dx$, we will integrate each term separately. The power rule for integration states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$. We apply this rule to each term of the polynomial and add a constant of integration $C$.

We need to find the indefinite integral of the polynomial $\int (2x^3 + 5x^2 - 3x + 2) \, dx$.

Using the power rule for integration, we integrate each term separately:

• For $2x^3$, the integral is $\frac{2}{4} x^4 = \frac{1}{2} x^4$.
• For $5x^2$, the integral is $\frac{5}{3} x^3$.
• For $-3x$, the integral is $-\frac{3}{2} x^2$.
• For $2$, the integral is $2x$.

Combining all the results, we have: $$\int (2x^3 + 5x^2 - 3x + 2) \, dx = \frac{1}{2} x^4 + \frac{5}{3} x^3 - \frac{3}{2} x^2 + 2x + C$$

Final Answer