Integration by Substitution Calculator
Reverse chain rule for integration.
Formula first
Overview
Integration by substitution is a formal method in calculus used to simplify the integration of composite functions by changing the variable of integration. It serves as the integral equivalent of the chain rule, transforming a complex integrand into a simpler form where the antiderivative is more easily recognized. By identifying a function and its derivative within the integrand, the variable is shifted to u, streamlining the calculation process.
Symbols
Variables
k = Coefficient k, n = Power n, a = Lower limit a, b = Upper limit b, I = Integral result
Apply it well
When To Use
When to use: Apply this method when the integrand contains a function and its derivative, typically in the form of a composite function. It is particularly useful when dealing with powers of polynomials, trigonometric identities, or exponential terms where the exponent is non-linear.
Why it matters: This technique is essential for solving complex differential equations found in physics, such as those governing planetary motion or electromagnetism. It allows scientists to solve integrals that are otherwise impossible to evaluate, providing a bridge between symbolic representations and numerical solutions.
Avoid these traps
Common Mistakes
- Not replacing dx with du terms.
- Leaving x's in u integral.
One free problem
Practice Problem
Evaluate the definite integral of 2x(x² + 1)² dx from x = 0 to x = 1.
Solve for:
Hint: Substitute u = x² + 1.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, James. Calculus: Early Transcendentals.
- Wikipedia: Integration by substitution
- Calculus: Early Transcendentals, 8th Edition by James Stewart
- University Physics with Modern Physics, 15th Edition by Young and Freedman
- Stewart, James. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2016.
- Standard curriculum — A-Level Pure Mathematics (Integration)