Integrate the function y = 3x$^2$ + 2x - 5 with respect to x.

To solve this integration problem, we use the Power Rule for Integration.

1. Analyze the question

We are asked to find the indefinite integral of the polynomial function:

2. Explain the concept

The Power Rule for integration states that for any term $ax^n$:

(Where $C$ is the constant of integration, and $n \neq -1$).

Essentially, you add 1 to the exponent and then divide by the new exponent. For a constant term (like $-5$), it is treated as $-5x^0$, so its integral becomes $-5x^1$.

3. Show the solution

Let's integrate each term of the function $3x^2 + 2x - 5$ individually:

• For $3x^2$: Add 1 to the exponent ($2+1=3$) and divide by 3: $$\frac{3x^3}{3} = x^3$$

• For $2x$ (which is $2x^1$): Add 1 to the exponent ($1+1=2$) and divide by 2: $$\frac{2x^2}{2} = x^2$$

• For $-5$ (which is $-5x^0$): Add 1 to the exponent ($0+1=1$) and divide by 1: $$\frac{-5x^1}{1} = -5x$$

• Don't forget the constant: Always add $+ C$ at the end of an indefinite integral.

Combining these, we get:

4. Identify the answer

Comparing our result to the given options, it matches Option A.

5. Exam Tip

When integrating, always perform a quick mental check by differentiating your answer. If you differentiate $x^3 + x^2 - 5x + C$, you get $3x^2 + 2x - 5$, which is the original function. If you get the original function back, you know your integration is 100% correct!