https://file.aiquickdraw.com/mathbot/file/1726900355467oz6ecj5t.jpg

The image displays a definite integral problem:

\[
\int_1^3 (4x - 5)^2 \, dx
\]

The task is to evaluate this integral and provide the answer in fraction form. Let's solve it step by step.

### Step 1: Expand the integrand \((4x - 5)^2\)

\[
(4x - 5)^2 = (4x)^2 - 2 \cdot 4x \cdot 5 + 5^2 = 16x^2 - 40x + 25
\]

### Step 2: Set up the integral

\[
\int_1^3 (16x^2 - 40x + 25) \, dx
\]

### Step 3: Integrate term by term

\[
\int (16x^2) \, dx = \frac{16x^3}{3}
\]
\[
\int (-40x) \, dx = -20x^2
\]
\[
\int (25) \, dx = 25x
\]

So, the integral becomes:

\[
\left[ \frac{16x^3}{3} - 20x^2 + 25x \right]_1^3
\]

### Step 4: Evaluate at the limits

Now, we evaluate this at \(x = 3\) and \(x = 1\) and subtract:

At \(x = 3\):

\[
\frac{16(3)^3}{3} - 20(3)^2 + 25(3) = \frac{16(27)}{3} - 20(9) + 75 = 144 - 180 + 75 = 39
\]

At \(x = 1\):

\[
\frac{16(1)^3}{3} - 20(1)^2 + 25(1) = \frac{16(1)}{3} - 20(1) + 25 = \frac{16}{3} - 20 + 25 = \frac{16}{3} + 5 = \frac{31}{3}
\]

### Step 5: Subtract the values

\[
39 - \frac{31}{3} = \frac{117}{3} - \frac{31}{3} = \frac{86}{3}
\]

Thus, the exact value of the integral is:

\[
\boxed{\frac{86}{3}}
\]

Would you like more details on any of the steps?

Here are five related questions you might consider:
1. How can you verify the result of a definite integral using numerical methods?
2. What are the key rules for expanding binomials like \((4x - 5)^2\)?
3. How does the fundamental theorem of calculus apply to solving definite integrals?
4. What are some common errors when performing polynomial integration?
5. How does changing the limits of integration affect the result?

**Tip:** When expanding a squared binomial like \((ax - b)^2\), always apply the formula \(a^2x^2 - 2abx + b^2\) to avoid mistakes.

Evaluate the definite integral: ∫_1^3 (4x - 5)^2 dx. Write an exact answer in fraction form.

Learn how to evaluate the definite integral of (4x - 5)^2 from 1 to 3 using polynomial expansion and the power rule for integration. This calculus problem covers the expansion of binomials and integration techniques, suitable for undergraduate students.

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