Step by Step
1
Recognize the pattern
U-substitution is essentially the chain rule run in reverse for integration. Look for an integral containing a composite function multiplied by the derivative of its inner function: ∫f(g(x))·g'(x)dx.
2
Choose u and find du
Let u equal the inner function g(x). Then du = g'(x)dx — the derivative of u with respect to x, times dx.
3
Replace everything and integrate
Substitute u for g(x) and du for g'(x)dx throughout the integral, leaving an integral purely in terms of u. Integrate this simpler expression.
4
Substitute back
Once you've integrated in terms of u, replace u with its original expression in x to give your final answer — don't leave the answer in terms of u.
Applied Walkthrough
1
Evaluate ∫2x·cos(x²)dx. Notice the pattern: cos(x²) is a composite function, and 2x is exactly the derivative of the inner function x².
2
Let u = x² (the inner function). Then du = 2x·dx (the derivative of the inner function, times dx) — which conveniently matches exactly what's left in the integral.
3
Substitute: the integral becomes ∫cos(u)du, which integrates to sin(u) + C.
4
Substitute back u = x²: the final answer is sin(x²) + C.
Exam Application
Exams test whether you can recognize when u-substitution applies (spotting a composite function multiplied by its inner derivative), correctly compute du, and remember to substitute back to the original variable at the end.
⚠ Common Trap
The most common trap is forgetting to substitute back to the original variable x at the end — leaving the final answer in terms of u is an incomplete answer, since the original integral was in terms of x.
✓ Quick Self-Check
1. What pattern signals that u-substitution should be used?
An integral containing a composite function multiplied by the derivative of its inner function: f(g(x))·g'(x).
Tap to reveal / hide
2. Evaluate ∫2x·cos(x²)dx using u-substitution.
sin(x²) + C.
Tap to reveal / hide
3. What is u-substitution essentially the reverse of?
The chain rule, run in reverse for integration.
Tap to reveal / hide
4. What is the final step of u-substitution that's often forgotten?
Substituting back from u to the original variable x.
Tap to reveal / hide
5. If u = x², what does du equal?
2x·dx.
Tap to reveal / hide