How do I use the chain rule when calculating an antiderivative?
To calculate the antiderivative of a function using the chain rule, you need to follow these steps:

Identify the independent and dependent variables. For example, if you are finding the derivative of f(x) = x^2 + 3x, then the independent variable is x and the dependent variable is y = f(x).
Draw the function graph. This will help you understand how the functions relate to each other.
Express the original function in terms of its components. For example, if you want to find ∫f'(x) dx, you would express f(x) as f(x) = x^2 + 2x + c, where c is an arbitrary constant.
Use the formula for the product of two derivatives. The formula for the product of two derivates is d/dx (uv') = u'dv' + v'du'. Apply this formula to your problem and simplify.
Substitute the resulting expression into the original integral formula. For example, if we apply the above rules to our previous problem, we get ∫f'(x) dx = (x^3 + 2x^2 + 6x + 10) / 5.
Simplify the result as necessary. For example, we can divide by 5 to remove the denominators from the integrand. Then, we can perform some basic algebra on the remaining expression to reduce it to a simpler form.
The final answer is the antiderivative of the original function. Note that the chain rule may not always be the easiest or most straightforward way to solve an integral; there may be alternative methods available that can be more efficient or easier to understand. However, the chain rule provides a general approach that works for many types of integrals.