Find ∫(x2 + 3x − 5)dx

To solve the indefinite integral $\int (x^2 + 3x - 5) \, dx$, we apply the Power Rule for Integration.

1. The Concept

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

(Where $C$ or $k$ is the constant of integration).

When integrating a polynomial, we integrate each term individually.

2. Step-by-Step Solution

We integrate each part of the expression $(x^2 + 3x - 5)$ separately:

• For $x^2$: Increase the power by 1 ($2+1=3$) and divide by the new power: $$\int x^2 \, dx = \frac{x^3}{3}$$

• For $3x$: The power of $x$ is 1. Increase it by 1 ($1+1=2$) and divide by the new power: $$\int 3x^1 \, dx = \frac{3x^2}{2}$$

• For $-5$: A constant $a$ integrates to $ax$. So, $-5$ becomes $-5x$: $$\int -5 \, dx = -5x$$

• Adding the constant: Since this is an indefinite integral, we add the constant of integration, $k$.

3. Combining the results

Putting it all together:

Comparing this result to your options, it matches option C.

Exam Tip

Always remember that integration is the reverse of differentiation. If you are ever unsure of your answer during an exam, quickly differentiate your result. If you differentiate $\frac{x^3}{3} + \frac{3x^2}{2} - 5x + k$, you will get back to the original expression $(x^2 + 3x - 5)$, confirming your answer is correct!