91Ó°ÊÓ

Integration by substitution is a method used to simplify complicated integrals. It's similar to the chain rule in differentiation, but it works for integration. In our problem, we have the integral \(\text{\int (5x - 7)^{-2} \, dx}\). This looks complicated, but there's a trick to simplify it by changing variables.

We set \(\text{u = 5x - 7}\) to transform the integral into a simpler form. When we substitute \(\text{u}\) for \(\text{5x - 7}\), we also need to change \(\text{dx}\) to \(\text{du}\) using the derivative of \(\text{u}\), which is \(\text{du/dx = 5}\). Therefore, \(\text{dx = du/5}\).

An antiderivative is a function that reverses differentiation. In other words, if you have a function and you differentiate it, then find a function whose derivative is the original function, that new function is the antiderivative. It's also referred to as the integral.

For example, in the integral \(\text{\int u^{-2} \, du}\), the antiderivative is \(\text{-u^{-1} + C}\). Therefore, when we transform back to the \(\text{x}\) form, we get the function that reverses the differentiation process of \(\text{ (5x - 7)^{-2}}\).

Solving calculus problems often involves several steps: understanding the problem, setting up the integral or differential equation, using suitable substitution or differentiation rules, and then verifying the result. Let's break down our problem:

• Understand the problem: We need to find a constant \(\text{k}\) that makes the integral equation true.
• Setup the integral: We start with \(\text{\int (5x - 7)^{-2} \, dx}\).
• Use the substitution \(\text{u = 5x - 7}\).
• Integrate with respect to \(\text{u}\).
• Compare our solution with the given format to solve for \(\text{k}\).
• Verify the solution by differentiating the antiderivative.

The chain rule is crucial when differentiating composite functions. It's used to find the derivative of \(\text{f(g(x))}\) by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

When we verify our integral result, we use the chain rule to differentiate \(\text{-\frac{1}{5}(5x - 7)^{-1} + C}\). Here's how:

• Outer function is \(\text{(5x - 7)^{-1}}\) and derivative is \(\text{-u^{-2}}\) where \(\text{u = 5x - 7}\).
• Inner function is \(\text{5x - 7}\) and derivative is \(\text{5}\).
• Combining, \(\text{\frac{d}{dx} ( -\frac{1}{5}(5x - 7)^{-1} ) = -\frac{1}{5} \cdot -1 \cdot (5x - 7)^{-2} \cdot 5 = (5x - 7)^{-2}}\).

This verification ensures that our antiderivative is correct and reaffirms the value of \(\text{k = -\frac{1}{5}}\).