MISSION · Calculus
Integration by Substitution
Renaming the awkward part until the integral becomes one you know
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
Substitution is the chain rule run backwards: spot an inner function and its derivative sitting together, rename the inner one \(u\), and the integral collapses into a standard one.
The pattern to look for
\[\int f(g(x)) \cdot g'(x)\,dx = \int f(u)\,du \quad \text{where } u = g(x)\]The essential feature is that the derivative of the inner function is already there, as a factor. That’s not a coincidence you hope for — it’s what differentiating a composite function leaves behind, so any integral that came from a chain rule will have it.
\[\int 2x\cos(x^2)\,dx\]The inner function is \(x^2\), and its derivative \(2x\) is right there multiplying. Let \(u = x^2\), so \(du = 2x\,dx\), and the whole thing becomes:
\[\int \cos u\,du = \sin u + C = \sin(x^2) + C\]The mechanics
- Choose \(u\) — usually the inside of a bracket, a root, an exponent, or a denominator.
- Differentiate: \(\frac{du}{dx} = g'(x)\), so \(du = g'(x)\,dx\).
- Replace everything. No \(x\) may survive into the new integral.
- Integrate in terms of \(u\).
- Substitute back — unless the integral is definite, in which case see below.
Step 3 is the test of whether your choice was right. If an \(x\) is left over that you can’t express in \(u\), the substitution has failed and you need a different \(u\) or a different method entirely.
Constants are adjustable
A missing constant factor is not a problem, because constants can be moved across the integral sign.
\[\int x\,e^{x^2}dx\]Here \(u = x^2\) gives \(du = 2x\,dx\), but there’s only one \(x\). Write \(x\,dx = \frac{1}{2}du\) and carry the half outside:
\[\frac{1}{2}\int e^u\,du = \frac{1}{2}e^{x^2} + C\]A missing variable factor, though, is fatal. \(\int e^{x^2}dx\) has no \(x\) available and cannot be done this way — or, as it happens, by any elementary method at all.
Definite integrals: change the limits
For \(\int_0^2 2x(x^2+1)^3dx\) with \(u = x^2 + 1\), the limits are \(x\) values and the new integral is in \(u\). Convert them: \(x = 0\) gives \(u = 1\), and \(x = 2\) gives \(u = 5\).
\[\int_1^5 u^3\,du = \left[\frac{u^4}{4}\right]_1^5 = \frac{625 - 1}{4} = 156\]Changing the limits saves substituting back, and it’s less error-prone. What you must never do is keep the original limits with the new variable — that silently integrates over the wrong interval.
Common mix-ups
- Leaving an \(x\) in the transformed integral.
- Forgetting to convert \(dx\) into \(du\). They differ by the factor \(g'(x)\).
- Keeping the old limits on a definite integral after substituting.
- Dropping \(+C\) on an indefinite one.
Check yourself
Find \(\int 6x^2(x^3 + 4)^5dx\).
Let \(u = x^3 + 4\), so \(du = 3x^2dx\) and \(6x^2dx = 2\,du\). The integral becomes \(2\int u^5du = \frac{u^6}{3} + C = \frac{(x^3+4)^6}{3} + C\).
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES