What is the result of ∫ (3x^2 + 5) dx?

To evaluate the integral ∫ (3x^2 + 5) dx, we apply the power rule of integration, which states that the integral of x^n is (x^(n+1))/(n+1) for any real number n, except when n = -1.

In this case, we break down the integral into two separate parts:

1. The integral of 3x^2:

Using the power rule, we find that:

∫ 3x^2 dx = 3 * (x^(2+1))/(2+1) = 3 * (x^3)/3 = x^3.

2. The integral of 5:

The integral of a constant is straightforward:

∫ 5 dx = 5x.

Combining these results gives us:

∫ (3x^2 + 5) dx = x^3 + 5x + C, where C is the integration constant.

Therefore, the correct answer captures the proper result of integrating the given function, including the integration constant. This aligns perfectly with the first choice provided.

Other options do not derive from the proper application of the integration rules, leading to incorrect formulations or missing necessary