Substitution (Change of Variable) Rule
The Substitution Rule, also known as the Change of Variable Rule, is a powerful technique used in calculus to simplify the integration of complex functions. It involves replacing the original variable of integration with a new variable to transform the integral into a more manageable form.
Mathematical Notation
If ∫ f(g(x)) g'(x) dx represents an integral involving the composition of functions and their derivatives, the Substitution Rule states that this integral can be rewritten as:
∫ f(u) du
where u = g(x).
Chain Rule in Mathematics
The Chain Rule is a fundamental concept in calculus that deals with the differentiation of composite functions. It allows us to find the derivative of a composition of two or more functions.
Mathematical Notation
If y = f(g(x)), where f and g are functions, the Chain Rule can be expressed mathematically as:
(f ∘ g)'(x) = f'(g(x)) * g'(x)
This rule states that the derivative of the composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.
How It Works
The Chain Rule is essential for calculating derivatives of functions with multiple layers of composition. It allows us to break down the differentiation process into smaller steps, making it possible to handle complex functions effectively.
Substitution (Change of Variable) Rule Examples
Example 1
Compute the integral:
∫ 3x * cos(x^2) dx
u = x^2, du = 2x dx
Making the substitution: ∫ 3x * cos(x^2) dx = (3/2) ∫ cos(u) du = (3/2) * sin(u) + C = (3/2) * sin(x^2) + C
Example 2
Compute the integral:
∫ x * e^(x^2) dx
u = x^2, du = 2x dx
Making the substitution: ∫ x * e^(x^2) dx = (1/2) ∫ e^u du = (1/2) * e^u + C = (1/2) * e^(x^2) + C
Example 3
Compute the integral:
∫ (x + 1)^3 dx
u = x + 1, du = dx
Making the substitution: ∫ (x + 1)^3 dx = ∫ u^3 du = (1/4) u^4 + C = (1/4) (x + 1)^4 + C
Example 4
Compute the integral:
∫ x * sin(x^2) dx
u = x^2, du = 2x dx
Making the substitution: ∫ x * sin(x^2) dx = (1/2) ∫ sin(u) du = -(1/2) * cos(u) + C = -(1/2) * cos(x^2) + C
Example 5
Compute the integral:
∫ x * ln(x) dx
u = ln(x), du = (1/x) dx
Making the substitution: ∫ x * ln(x) dx = ∫ u du = (1/2) u^2 + C = (1/2) ln^2(x) + C