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Integration by Substitution (Reverse Chain Rule):
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Integration by substitution, also known as the reverse chain rule, is a method for finding integrals that are in the form of the product of a composite function and the derivative of its inner function. It's essentially the reverse process of the chain rule in differentiation.
The calculator uses the integration by substitution formula:
Where:
Explanation: The method works by recognizing when an integral contains both a function and its derivative, allowing for substitution to simplify the integral.
Details: The reverse chain rule is fundamental for solving integrals of composite functions. It's widely used in calculus for solving problems in physics, engineering, and economics where functions are often composed of other functions.
Tips: Enter the composite function (f(g(x))) and its derivative (g'(x)). The calculator will attempt to find the antiderivative using substitution. For best results, ensure your function is in the correct form for substitution.
Q1: When should I use integration by substitution?
A: Use it when you can identify a function and its derivative within the integral, typically in the form ∫f(g(x))·g'(x)dx.
Q2: What if I can't find a function and its derivative?
A: You may need to use other integration techniques like integration by parts, partial fractions, or trigonometric substitution.
Q3: How do I choose the substitution?
A: Look for the inner function (g(x)) whose derivative (g'(x)) appears as a factor in the integrand.
Q4: What are common substitution examples?
A: Common substitutions include u = x², u = sin(x), u = ln(x), or any composite function whose derivative is present.
Q5: Does this work for definite integrals?
A: Yes, but you must either change the limits of integration according to your substitution or convert back to the original variable before evaluating.