Integration by U-Substitution

Simplify integration using coordinate transformations. By substituting $u = g(x)$ and $du = g'(x)dx$, complex composite integrals transform into simpler elementary forms with matching areas.

Interactive Sandbox

x-space vs u-space Area Equivalence

Select an integration preset. Adjust the upper boundary limit slider ($b$). Observe how the coordinate bounds and height stretch in $u$-space, while keeping the total integrated area (shaded region) identical in both spaces.

Original: ∫ f(g(x)) * g'(x) dx (x-space)
Transformed: ∫ f(u) du (u-space)

U-Substitution Theorem

∫ab f(g(x)) • g'(x) dx = ∫g(a)g(b) f(u) du
Upper Bound Limit (b) 1.50
Changing the limits dynamically morphs the integration ranges in both coordinates, preserving the exact area values.
Algebraic Solver

Substitution Steps

Trace the differential substitutions and limits mapping from $x$ to $u$.

U-Substitution Mapping Steps


Substitution Values


Original bounds [a, b]

x in [0.00, 1.50]

Transformed bounds [g(a), g(b)]

u in [0.00, 2.25]

Integrated Area Result

Area = 0.7781

Evaluation

Substitution Quiz

Test your understanding of coordinate limits conversion and differential substitutions.

Question 1 of 5 Score: 0/0

Evaluate the integral ∫ 2x e^(x²) dx. What is the best choice for u?

Correct Answer!

Explanation text...

Substitution Rules


  • Choose u = g(x) (the inner composite function)
  • Find du = g'(x) dx (differential matching)
  • Map bounds: u_lower = g(a) | u_upper = g(b)