Find the integral of (2x^3 - 6x + 3) dx.

Find the integral of (2x^3 - 6x + 3) dx.

• A. (1/2)x^4 - 3x^2 + 3x + C
• B. (1/4)x^4 - 3x^2 + 3 + C
• C. (1/2)x^4 - 3x + C
• D. 2x^4 - 3x^2 + 3 + C

Answer: (A)

To find the integral of the function \(2x^3 - 6x + 3\), we apply the power rule of integration to each term individually. The power rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. 1. For the term \(2x^3\): - The integral is \(\int 2x^3 \, dx = 2 \cdot \frac{x^{4}}{4} = \frac{1}{2}x^4\). 2. For the term \(-6x\): - The integral is \(\int -6x \, dx = -6 \cdot \frac{x^{2}}{2} = -3x^2\). 3. For the constant term \(3\): - The integral is \(\int 3 \, dx = 3x\). Combining these results gives: \[ \int (2x^3 - 6x + 3) \, dx = \frac{1}{2}x^4 - 3

To find the integral of the function (2x^3 - 6x + 3), we apply the power rule of integration to each term individually. The power rule states that the integral of (x^n) is (\frac{x^{n+1}}{n+1} + C), where (C) is the constant of integration.

1. For the term (2x^3):

• The integral is (\int 2x^3 , dx = 2 \cdot \frac{x^{4}}{4} = \frac{1}{2}x^4).

2. For the term (-6x):

• The integral is (\int -6x , dx = -6 \cdot \frac{x^{2}}{2} = -3x^2).

3. For the constant term (3):

• The integral is (\int 3 , dx = 3x).

Combining these results gives:

[

\int (2x^3 - 6x + 3) , dx = \frac{1}{2}x^4 - 3