What is the result of the integral ∫ (e^x + 2)e^x dx?

What is the result of the integral ∫ (e^x + 2)e^x dx?

• A. (1/2)e^(2x) + e^x + C
• B. (1/2)e^(2x) + 2e^x + C
• C. (1/3)e^(2x) + 2e^x + C
• D. (1/2)e^(2x) + 3e^x + C

Answer: (B)

To evaluate the integral ∫ (e^x + 2)e^x dx, we can first simplify the integrand. This can be rewritten as: ∫ (e^x * e^x + 2e^x) dx, which simplifies to ∫ (e^(2x) + 2e^x) dx. Now, we can break this integral into two separate integrals: ∫ e^(2x) dx + ∫ 2e^x dx. The first integral, ∫ e^(2x) dx, can be solved by using the formula for the integral of an exponential function. To integrate e^(2x), we use the fact that ∫ e^(ax) dx = (1/a)e^(ax) + C. Here, a = 2, so we get: ∫ e^(2x) dx = (1/2)e^(2x) + C. For the second integral, ∫ 2e^x dx, we again use the formula for the exponential function and multiply by 2: ∫ 2e^x dx = 2 * (e^x) + C = 2e^x + C

To evaluate the integral ∫ (e^x + 2)e^x dx, we can first simplify the integrand. This can be rewritten as:

∫ (e^x * e^x + 2e^x) dx, which simplifies to ∫ (e^(2x) + 2e^x) dx.

Now, we can break this integral into two separate integrals:

∫ e^(2x) dx + ∫ 2e^x dx.

The first integral, ∫ e^(2x) dx, can be solved by using the formula for the integral of an exponential function. To integrate e^(2x), we use the fact that

∫ e^(ax) dx = (1/a)e^(ax) + C.

Here, a = 2, so we get:

∫ e^(2x) dx = (1/2)e^(2x) + C.

For the second integral, ∫ 2e^x dx, we again use the formula for the exponential function and multiply by 2:

∫ 2e^x dx = 2 * (e^x) + C = 2e^x + C