Sure, I'd be happy to explain the rules of integration and how we arrived at the solution.

The process of integration involves finding the antiderivative of a function. In other words, if f(x) is a function, then its antiderivative F(x) is a function such that F'(x) = f(x), where F'(x) denotes the derivative of F(x) with respect to x.

There are several rules of integration that we use to find antiderivatives of functions. These rules include the power rule, the constant multiple rule, the sum rule, and the substitution rule.

In the case of the function 5x^2, we can use the power rule of integration to find its antiderivative. The power rule states that if f(x) = x^n, where n is any constant except for -1, then the antiderivative of f(x) is F(x) = (1/(n+1)) x^(n+1) + C, where C is the constant of integration.

So for the function 5x^2, we can rewrite it as f(x) = 5x^(2). Using the power rule, we get:

∫ 5x^2 dx = 5 ∫ x^2 dx

Then, using the power rule again, we can integrate x^2 as follows:

∫ x^2 dx = (1/3) x^3 + C

Substituting this expression back into the original equation, we get:

∫ 5x^2 dx = 5 ∫ x^2 dx = 5 (1/3) x^3 + C

Simplifying this expression, we get:

∫ 5x^2 dx = (5/3) x^3 + C

Therefore, the antiderivative of 5x^2 is (5/3) x^3 plus a constant C.
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