To find the integral of the function [tex]\( 12x^2 - 7x + 3 \)[/tex] with respect to [tex]\( x \)[/tex], we need to integrate each term of the polynomial separately.
We start with the function:
[tex]\[ \int (12x^2 - 7x + 3) \, dx \][/tex]
We can split this integral into three separate integrals:
[tex]\[ \int 12x^2 \, dx - \int 7x \, dx + \int 3 \, dx \][/tex]
Now, we will integrate each term individually.
1. Integral of [tex]\( 12x^2 \)[/tex]:
[tex]\[ \int 12x^2 \, dx \][/tex]
To integrate [tex]\( 12x^2 \)[/tex], we use the power rule for integration, which states:
[tex]\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} \][/tex]
For [tex]\( 12x^2 \)[/tex], [tex]\( n = 2 \)[/tex]:
[tex]\[ \int 12x^2 \, dx = 12 \cdot \frac{x^{2+1}}{2+1} = 12 \cdot \frac{x^3}{3} = 4x^3 \][/tex]
2. Integral of [tex]\( 7x \)[/tex]:
[tex]\[ \int -7x \, dx \][/tex]
Again, we use the power rule. For [tex]\( -7x \)[/tex], [tex]\( n = 1 \)[/tex]:
[tex]\[ \int -7x \, dx = -7 \cdot \frac{x^{1+1}}{1+1} = -7 \cdot \frac{x^2}{2} = -\frac{7x^2}{2} \][/tex]
3. Integral of the constant 3:
[tex]\[ \int 3 \, dx \][/tex]
The integral of a constant [tex]\( c \)[/tex] is:
[tex]\[ \int c \, dx = cx \][/tex]
So for the constant 3:
[tex]\[ \int 3 \, dx = 3x \][/tex]
Now we combine the results of these integrals:
[tex]\[ \int (12x^2 - 7x + 3) \, dx = 4x^3 - \frac{7x^2}{2} + 3x \][/tex]
Therefore, the integral of [tex]\( 12x^2 - 7x + 3 \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ \int (12x^2 - 7x + 3) \, dx = 4x^3 - \frac{7x^2}{2} + 3x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
