Integration by Substitution
Reverse the chain rule by renaming an inner function, changing the differential and, for a definite integral, changing the limits with it.
What a learner can do afterwards
- Choose u so that du absorbs the leftover factor in the integrand
- Convert the limits of a definite integral when substituting
- Say why a substitution fails when the required factor is missing
1 · Read
Some integrands come from the chain rule. You can see an inner expression g(x) with its derivative sitting beside it. Call the inner part u, so du = g'(x) dx. Then rewrite everything in u and integrate there.
Take the integral of 2x cos(x^2) dx. The inner expression is x^2, so set u = x^2 and du = 2x dx. The integral becomes the integral of cos(u) du, which is sin(u) + C. Swap back to get sin(x^2) + C. The same move turns x e^(x^2) dx into (1/2) e^(x^2) + C, and u = ln x turns (ln x)/x dx into u du.
For a definite integral, carry the limits with you. Either convert them with u = g(x), so x from 0 to 1 may become u from 1 to 2, or swap back to x before you evaluate. Never evaluate x limits against a u integrand.
Substitution fails when the needed factor is missing. With u = x^2, du = 2x dx needs an x, so the integral of cos(x^2) dx cannot be fixed this way. Likewise e^(x^2) has no elementary antiderivative through substitution. Always check that du is really there.
Name the inner expression u, carry du with you, and change the limits when they exist.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.