Certainly! Let's find the integral of the function [tex]\( 5x^2 - 3x + 7 \)[/tex] with respect to [tex]\( x \)[/tex].
We start by integrating each term of the polynomial separately.
1. Integrate [tex]\( 5x^2 \)[/tex]:
To integrate [tex]\( 5x^2 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\int 5x^2 \, dx = 5 \int x^2 \, dx
\][/tex]
Recall that the integral of [tex]\( x^n \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( \frac{x^{n+1}}{n+1} \)[/tex]. For [tex]\( x^2 \)[/tex], [tex]\( n = 2 \)[/tex]:
[tex]\[
\int x^2 \, dx = \frac{x^{3}}{3}
\][/tex]
Therefore:
[tex]\[
5 \int x^2 \, dx = 5 \cdot \frac{x^{3}}{3} = \frac{5x^3}{3}
\][/tex]
2. Integrate [tex]\( -3x \)[/tex]:
To integrate [tex]\( -3x \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\int -3x \, dx = -3 \int x \, dx
\][/tex]
Recall that the integral of [tex]\( x \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( \frac{x^2}{2} \)[/tex]:
[tex]\[
\int x \, dx = \frac{x^2}{2}
\][/tex]
Therefore:
[tex]\[
-3 \int x \, dx = -3 \cdot \frac{x^2}{2} = -\frac{3x^2}{2}
\][/tex]
3. Integrate [tex]\( 7 \)[/tex]:
To integrate the constant [tex]\( 7 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\int 7 \, dx = 7 \int 1 \, dx = 7x
\][/tex]
Now, we combine these results together:
[tex]\[
\int (5x^2 - 3x + 7) \, dx = \frac{5x^3}{3} - \frac{3x^2}{2} + 7x
\][/tex]
Thus, the integral is:
[tex]\[
\frac{5x^3}{3} - \frac{3x^2}{2} + 7x + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
