Home

Integration by Substitution Online Calculator

U-substitution verifier. Built-in examples numerically confirm ∫f(g(x))g′(x)dx is unchanged after substitution, with steps—for u-sub practice.

Show more ▾
∫ f(g(x))g′(x)dx = ∫ f(u)du
Choose a preset to fill in f(x) and interval.
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Integration by Substitution

1. Definition
When the integrand has the form f(g(x))·g′(x), set u=g(x) so du=g′(x)dx. The integral becomes ∫ f(u)du, then substitute back. This is u-substitution.
2. Symbols
SymbolsMeaning
u=g(x)Inner function
du=g′(x)dx
∫f(u)duNew integral after substitution
a,bOriginal limits
g(a),g(b)New limits
3. How it works
  • Choose a classic composite example;
  • Adaptive Simpson directly integrates the original ∫_a^b f(x)dx;
  • The value matches the analytic result after substitution, verifying the method.
4. Steps
  1. Pick a substitution example from the dropdown;
  2. Fields fill automatically with a note on u=g(x);
  3. Click Calculate and compare the numeric and analytic values.
5. Example
Example: ∫₀¹ 2x·cos(x²)dx. Let u=x², du=2x dx, limits 0→1. Then ∫₀¹ cos u du=[sin u]₀¹=sin1≈0.8415. Tool: 0.841471.
6. Pitfalls
Change the limits when you substitute;
Do not forget the coefficient, e.g. du=2x dx not x dx;
For definite integrals, just use the new limits—no back-substitution needed.

❓ FAQ | Integration by Substitution

What is the key to substitution?
Spot the structure outer(inner) × (inner)′; set the inner function to u and complete du.
Why no back-substitution for definite integrals?
The limits become u-limits, so ∫_{g(a)}^{g(b)} f(u)du gives the number directly.
Does the tool auto-substitute?
No; it offers classic examples and verifies numerically that substitution preserves the value.
How do the limits change?
x=a maps to u=g(a), x=b to u=g(b).
Which functions are supported?
+−*/^, sin cos exp log sqrt abs etc.; never omit *.