https://file.aiquickdraw.com/mathbot/file/1736919027532bseqngmg.jpg

We are given the integral:

\[
\int (a + 1)(a - 2) \, da
\]

Let's solve it step by step:

### Step 1: Expand the expression inside the integral
First, expand the product \((a + 1)(a - 2)\):

\[
(a + 1)(a - 2) = a^2 - 2a + a - 2 = a^2 - a - 2
\]

Now, the integral becomes:

\[
\int (a^2 - a - 2) \, da
\]

### Step 2: Integrate term by term
We apply the power rule of integration, which states:

\[
\int a^n \, da = \frac{a^{n+1}}{n+1} \quad \text{for} \quad n \neq -1
\]

So, integrating each term:

\[
\int a^2 \, da = \frac{a^3}{3}, \quad \int (-a) \, da = -\frac{a^2}{2}, \quad \int (-2) \, da = -2a
\]

### Step 3: Combine the results
\[
\int (a^2 - a - 2) \, da = \frac{a^3}{3} - \frac{a^2}{2} - 2a + C
\]

where \(C\) is the constant of integration.

### Final Answer
\[
\int (a + 1)(a - 2) \, da = \frac{a^3}{3} - \frac{a^2}{2} - 2a + C
\]

Would you like any further clarification? Let me know if you have any questions!

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Here are five related questions you might find helpful:
1. How do you handle integration by parts, and when is it used?
2. What are the rules for integrating trigonometric functions?
3. How can partial fractions be used in integration?
4. What is the difference between definite and indefinite integrals?
5. How do you check your integration results using differentiation?

**Tip**: Always simplify the expression inside an integral before applying rules to make your work easier!

Evaluate the integral ∫(a + 1)(a - 2) da.

Learn how to solve the integral ∫(a + 1)(a - 2) da step by step. This solution covers polynomial expansion, the power rule for integration, and term-by-term integration techniques. Ideal for students in Grades 11-12.

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