Answer
In the integration by substitution process, we are essentially applying the chain rule backward to simplify the integral by making a suitable substitution.
Work Step by Step
Integration by substitution is closely related to the chain rule in differentiation. The chain rule states that if you have a composite function, say \( F(g(x)) \), then the derivative is given by the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.
Mathematically,
\[ \frac{d}{dx} F(g(x)) = F'(g(x)) \cdot g'(x) \]
And Now, when you perform integration by substitution, you are essentially working in reverse. You make a substitution to simplify the integral, and this substitution often involves a composite function. The chain rule comes into play when you differentiate the substitution.
So, in the integration by substitution process, you are essentially applying the chain rule backward to simplify the integral by making a suitable substitution.