Compute the value of 2 3
1 (8 2) d x x−
 . Hence, find the value of 2 3
1 2(8 2) d x x−

To solve the integral \( I = \int_1^2 (8x - 2) \, dx \), let's go step-by-step.

### Step 1: Solve \( \int_1^2 (8x - 2) \, dx \)

The given integral is:

\[
I = \int_1^2 (8x - 2) \, dx
\]

We can split the integral:

\[
I = \int_1^2 8x \, dx - \int_1^2 2 \, dx
\]

### Step 2: Evaluate \( \int_1^2 8x \, dx \)

The antiderivative of \( 8x \) is \( 4x^2 \). So, we have:

\[
\int_1^2 8x \, dx = \left[ 4x^2 \right]_1^2
\]
\[
= 4(2^2) - 4(1^2) = 4(4) - 4(1) = 16 - 4 = 12
\]

### Step 3: Evaluate \( \int_1^2 2 \, dx \)

The antiderivative of \( 2 \) is \( 2x \). So, we have:

\[
\int_1^2 2 \, dx = \left[ 2x \right]_1^2
\]
\[
= 2(2) - 2(1) = 4 - 2 = 2
\]

### Step 4: Combine the results

Now, subtract the two results:

\[
I = 12 - 2 = 10
\]

Thus, the value of \( \int_1^2 (8x - 2) \, dx \) is 10.

### Step 5: Find \( \int_1^2 2(8x - 2) \, dx \)

We are now asked to find the value of \( \int_1^2 2(8x - 2) \, dx \). Notice that this is simply twice the previous result:

\[
\int_1^2 2(8x - 2) \, dx = 2 \times \int_1^2 (8x - 2) \, dx = 2 \times 10 = 20
\]

### Final Answer:

- The value of \( \int_1^2 (8x - 2) \, dx = 10 \)
- The value of \( \int_1^2 2(8x - 2) \, dx = 20 \)

Would you like any details on the steps or have any questions?

### Related Questions:
1. How do you find the antiderivative of a linear function?
2. What is the fundamental theorem of calculus?
3. How do you split an integral into simpler parts?
4. How would the solution change if the limits of integration were different?
5. Can you apply the same method to a definite integral with a more complex function?

### Tip:
Always check if the integral can be simplified or factored out to make the calculation easier before solving

Compute the value of ∫₁² (8x - 2) dx. Hence, find the value of ∫₁² 2(8x - 2) dx.

Learn how to compute the value of ∫₁² (8x - 2) dx and its related integral ∫₁² 2(8x - 2) dx with detailed step-by-step instructions. This problem explores key calculus concepts such as definite integrals and the Fundamental Theorem of Calculus. Ideal for students in undergraduate calculus courses.

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