Answer :
Let's integrate the function [tex]\( k(x) = 4x^3 + 2x^2 - 5 \)[/tex].
1. Identify the function and its terms:
[tex]\[ k(x) = 4x^3 + 2x^2 - 5 \][/tex]
2. Integrate each term separately:
- For the term [tex]\( 4x^3 \)[/tex]:
[tex]\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{x^4}{4} \right) = x^4 \][/tex]
- For the term [tex]\( 2x^2 \)[/tex]:
[tex]\[ \int 2x^2 \, dx = 2 \int x^2 \, dx = 2 \left( \frac{x^3}{3} \right) = \frac{2x^3}{3} \][/tex]
- For the constant term [tex]\( -5 \)[/tex]:
[tex]\[ \int -5 \, dx = -5x \][/tex]
3. Combine the results of the integrals:
[tex]\[ \int (4x^3 + 2x^2 - 5) \, dx = x^4 + \frac{2x^3}{3} - 5x \][/tex]
4. Add the constant of integration (C):
[tex]\[ \int (4x^3 + 2x^2 - 5) \, dx = x^4 + \frac{2x^3}{3} - 5x + C \][/tex]
Therefore, the integral of [tex]\( k(x) = 4x^3 + 2x^2 - 5 \)[/tex] is:
[tex]\[ x^4 + \frac{2x^3}{3} - 5x + C \][/tex]
1. Identify the function and its terms:
[tex]\[ k(x) = 4x^3 + 2x^2 - 5 \][/tex]
2. Integrate each term separately:
- For the term [tex]\( 4x^3 \)[/tex]:
[tex]\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{x^4}{4} \right) = x^4 \][/tex]
- For the term [tex]\( 2x^2 \)[/tex]:
[tex]\[ \int 2x^2 \, dx = 2 \int x^2 \, dx = 2 \left( \frac{x^3}{3} \right) = \frac{2x^3}{3} \][/tex]
- For the constant term [tex]\( -5 \)[/tex]:
[tex]\[ \int -5 \, dx = -5x \][/tex]
3. Combine the results of the integrals:
[tex]\[ \int (4x^3 + 2x^2 - 5) \, dx = x^4 + \frac{2x^3}{3} - 5x \][/tex]
4. Add the constant of integration (C):
[tex]\[ \int (4x^3 + 2x^2 - 5) \, dx = x^4 + \frac{2x^3}{3} - 5x + C \][/tex]
Therefore, the integral of [tex]\( k(x) = 4x^3 + 2x^2 - 5 \)[/tex] is:
[tex]\[ x^4 + \frac{2x^3}{3} - 5x + C \][/tex]
