Step 1: Understanding the Concept:
Integration is the mathematical operation that functions as the inverse of differentiation. For a polynomial function, we can determine the indefinite integral by integrating each term independently. This relies on the linearity properties of calculus along with the fundamental power rule for integration.
Step 2: Key Formula or Approach:
The fundamental power rule for integration states that for any real power $n \neq -1$:
$$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $$
Additionally, the integral of any constant $k$ is $\int k \, dx = kx + C$, and the integration operator can be distributed across addition or subtraction:
$$ \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx $$
Step 3: Detailed Explanation:
Let's apply these integration integration rules to our integrand expression $\int (3x^2 + 4x - 5) \, dx$ term-by-term:
1. First term ($3x^2$): Factor out the constant coefficient and raise the power from $2$ to $3$:
$$ \int 3x^2 \, dx = 3 \cdot \left( \frac{x^{2+1}}{2+1} \right) = 3 \cdot \left( \frac{x^3}{3} \right) = x^3 $$
2. Second term ($4x$): Factor out the coefficient and raise the implicit power from $1$ to $2$:
$$ \int 4x \, dx = 4 \cdot \left( \frac{x^{1+1}}{1+1} \right) = 4 \cdot \left( \frac{x^2}{2} \right) = 2x^2 $$
3. Third term (constant $-5$): Integrating a constant append the independent variable $x$:
$$ \int -5 \, dx = -5x $$
Combining all three simplified components together and appending the mandatory arbitrary constant of integration ($C$), we arrive at the final integrated expression:
$$ \int (3x^2 + 4x - 5) \, dx = x^3 + 2x^2 - 5x + C $$
This derivation matches option (A).
Step 4: Final Answer:
The evaluated integral is $x^3 + 2x^2 - 5x + C$.