Integration by substitution
Replace a tangled part with a new letter u so the integral becomes simple. Change dx into du too, and change limits if the integral is definite.
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Analogy
Renaming a long word to one letter while you work, then translating back.
Example
Let u = x^2 + 1, so du = 2x dx.Integral of u^3 du = u^4/4.Back: (x^2 + 1)^4 / 4 + C.Differentiate to check.
In practice
The reverse of the chain rule. Look for a function whose derivative also appears.
Practice
Flashcards
- What substitution suits the integral of 2x (x^2 + 1)^3 dx?u = x^2 + 1, so du = 2x dx.
Questions people ask
How do I choose u in a substitution?
Choose the inside of a bracket, root or exponent, especially if its derivative appears elsewhere in the integral. In 2x times (x squared + 1) cubed, u = x squared + 1 works because du = 2x dx matches the extra factor exactly.
What happens to the limits in a definite integral?
If you change variable, change the limits to match. For u = x squared + 1 with x from 0 to 1, the new limits are u = 1 and u = 2. Then you do not need to switch back to x at the end.