Question

Evaluate: \[ \int (3x^2 + 4x - 5) \, dx \]

• A. \( (x^3 + 2x^2 - 5x + C) \)
• B. \( (3x^3 + 4x^2 - 5x + C) \)
• C. \( (x^2 + 2x - 5 + C) \)
• D. \( (x^3 + 4x^2 - 5 + C) \)

Hint:

When integrating polynomials on competitive exams, quickly differentiate the options in your head.
Taking the derivative of \( x^3 + 2x^2 - 5x + C \) instantly gives \( 3x^2 + 4x - 5 \).
This reverse verification technique is often faster and less prone to calculation errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In this problem, we are asked to find the indefinite integral of the algebraic polynomial function \( 3x^2 + 4x - 5 \).
Evaluating an indefinite integral requires finding the general antiderivative of the given function.
We must apply standard rules of integration to each term and append a constant of integration \( C \).

Step 2: Key Formula or Approach:
We use the fundamental rules of indefinite integration, which include:
1. The sum and difference rule: \( \int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx \).
2. The constant multiple rule: \( \int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx \).
3. The power rule of integration: For any real number \( n \neq -1 \), we have \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).
4. The integral of a constant: \( \int k \, dx = kx \).

Step 3: Detailed Explanation:
Let us apply the sum and difference rule to decompose the given integral into three simpler integrals:
\[ \int (3x^2 + 4x - 5) \, dx = \int 3x^2 \, dx + \int 4x \, dx - \int 5 \, dx \]
Now, we apply the constant multiple rule to factor out the numerical constants:
\[ = 3 \int x^2 \, dx + 4 \int x^1 \, dx - 5 \int 1 \, dx \]
Next, we apply the power rule to integrate each individual term with respect to \( x \):
For the first term, integrating \( x^2 \) yields:
\[ 3 \left( \frac{x^{2+1}}{2+1} \right) = 3 \left( \frac{x^3}{3} \right) = x^3 \]
For the second term, integrating \( x^1 \) yields:
\[ 4 \left( \frac{x^{1+1}}{1+1} \right) = 4 \left( \frac{x^2}{2} \right) = 2x^2 \]
For the third term, integrating the constant \( 5 \) yields:
\[ 5x \]
Now we combine all these results together and introduce the arbitrary constant of integration \( C \):
\[ \int (3x^2 + 4x - 5) \, dx = x^3 + 2x^2 - 5x + C \]
Let us verify this result by differentiating our antiderivative:
\[ \frac{d}{dx} (x^3 + 2x^2 - 5x + C) = 3x^2 + 4x - 5 \]
Since the derivative of our result matches the original integrand, the integration is verified.
Let us analyze why the other options are incorrect:
- Option (B) keeps the original coefficients instead of dividing them by the new powers.
- Option (C) represents differentiation rather than integration.
- Option (D) incorrectly performs the power rule on the second and third terms.

Step 4: Final Answer:
Therefore, the correct evaluation of the given integral matches Option (A).