Compute the integral ∫ (2x + 1)(x - 3) dx. What is the answer?

Compute the integral ∫ (2x + 1)(x - 3) dx. What is the answer?

• A. (1/3)x^3 - (7/2)x^2 + x + C
• B. (1/3)x^3 + (4/3)x^2 - 6x + C
• C. (2/3)x^3 - 3x^2 + C
• D. (1/2)x^3 - 2x^2 + 2x + C

Answer: (A)

To compute the integral ∫ (2x + 1)(x - 3) dx, start by expanding the integrand. The expression (2x + 1)(x - 3) simplifies to: \[ (2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3. \] Now, you can rewrite the integral as: \[ \int (2x^2 - 5x - 3) dx. \] Next, integrate each term separately. The integral of \(2x^2\) is \(\frac{2}{3}x^3\), the integral of \(-5x\) is \(-\frac{5}{2}x^2\), and the integral of \(-3\) is \(-3x\). So you get: \[ \int (2x^2 - 5x - 3) dx = \frac{2}{3}x^3 - \frac{5}{2}x^2 - 3x + C. \] Finally, we can combine these terms

To compute the integral ∫ (2x + 1)(x - 3) dx, start by expanding the integrand. The expression (2x + 1)(x - 3) simplifies to:

[

(2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3.

]

Now, you can rewrite the integral as:

[

\int (2x^2 - 5x - 3) dx.

]

Next, integrate each term separately. The integral of (2x^2) is (\frac{2}{3}x^3), the integral of (-5x) is (-\frac{5}{2}x^2), and the integral of (-3) is (-3x). So you get:

[

\int (2x^2 - 5x - 3) dx = \frac{2}{3}x^3 - \frac{5}{2}x^2 - 3x + C.

]

Finally, we can combine these terms