Integration by Substitution explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Integration by substitution is a method for evaluating integrals with an inside function.
It is often called the reverse of the chain rule.
This method is used in calculus, physics, area problems, probability, and differential equations.
Substitution changes a difficult integral into a simpler one.
Choose u as the inner function or repeated expression.
Differentiate u to connect du with dx.
Rewrite the whole integral in terms of u.
For definite integrals, either change the limits to alues or substitute back before using imits.
Meaning: u is the expression used to simplify the integral.
Definition: Choose u as the inside function when its derivative is also present.
For integral 2 dx, choose .
Meaning: du connects the new variable to dx.
Definition: If , then d'(x) dx.
This step lets you replace part of the integral.
Meaning: An integral without limits.
Definition: After integrating in u, substitute back in terms of x.
The final answer should usually use x again.
Meaning: An integral with limits.
Definition: Change limits to alues or substitute back before applying original limits.
Both approaches can work if used consistently.
Easy
Question: For integral 2 dx, choose u.
Medium
Question: Evaluate integral 2 dx.
Exam-level
Question: Evaluate integral from 0 to 1 of 2x() dx.
Leaving x terms after switching to u
Why it happens: Only part of the integral is replaced.
Correct approach: Rewrite the whole integral in one variable.
Forgettin
Why it happens: Indefinite integral habit is incomplete.
Correct approach: Ad for indefinite integrals.
Using old limits after changing to u
Why it happens: Limits are not converted.
Correct approach: Change limits or substitute back before applying them.
Why is substitution called reverse chain rule?
Because it undoes the pattern created by differentiating a composite function.
Can I choose any u?
A useful u should simplify the integral and match a derivative in the expression.
What happens to limits?
Change them to alues or return to x before using the original limits.
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