Integration by Substitution Calculator
Verified Calculation Engine
The online Integration by Substitution Calculator helps you calculate instantly and solve problems related to Integration. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Applies substitution method: ∫f(g(x))g'(x)dx = ∫f(u)du where u = g(x).
Substitution method:
Let u = g(x), then du = g'(x)dx
∫f(g(x))g'(x)dx = ∫f(u)du
After integration, substitute back u = g(x)
Substitution simplifies integration by changing variable
Inputs
Results
Worked Examples
Example 1: Basic
Find ∫2x(x² + 1)³dx using substitution
- function: 2x(x^2 + 1)^3
- integral: (x² + 1)⁴/4 + C
- substitutionUsed: u = x² + 1
- steps: Let u = x² + 1, du = 2x dx, then ∫u³du = u⁴/4 + C
Example 2: Board Level
Find ∫x·e^(x²)dx
- function: x*e^(x^2)
- integral: e^(x²)/2 + C
- substitutionUsed: u = x²
- steps: Let u = x², du = 2x dx, then ∫(1/2)e^u du = e^u/2 + C
About this calculator
Overview
Integration by Substitution Calculator evaluates Integration by Substitution for the math / integration library. Primary input cue: Function f(x) [required]: Function to integrate. Method anchor: Substitution method: Let u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du After integration, substitute back u = g(x). Substitution simplifies integration .
When to use
Use Integration by Substitution Calculator when you already know the inputs and need integration by substitution without setting up a spreadsheet.
Inputs explained
- Function f(x) [required]: Function to integrate
- Substitution u (Optional) [optional]: Substitution u = g(x) (if not provided, will suggest)
Formula / method
Substitution method: Let u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du After integration, substitute back u = g(x). Substitution simplifies integration by changing variable (calculator id: integration-by-substitution-calculator).
Worked example
Worked values: function = 2x(x^2 + 1)^3 → integral = (x² + 1)⁴/4 + C, substitutionUsed = u = x² + 1, steps = Let u = x² + 1, du = 2x dx, then ∫u³du = u⁴/4 + C
Interpreting results
If integration by substitution looks implausible, re-check units and which field is optional vs required.
Assumptions
- The wired execution function for `integration-by-substitution-calculator` is the source of numerical behavior.
- Inputs for `integration-by-substitution-calculator` follow the on-page labels; values/shapes must match this operation.
- Real-number arithmetic is used unless integration by substitution explicitly needs integers.
Limitations
- Display rounding can differ slightly from a CAS/symbolic exact form of integration by substitution.
- Edge cases outside the intended domain of integration by substitution may error or look unstable—treat them as out of scope.
- Complex/branch cuts and undefined points are not always interactively annotated for integration by substitution.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Substitution method: Let u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du After integration, substitute back u = g(x). Substitution simplifies integration by changing variable (calculator id: integration-by-substitution-calculator).
Example Calculation
Worked values: function = 2x(x^2 + 1)^3 → integral = (x² + 1)⁴/4 + C, substitutionUsed = u = x² + 1, steps = Let u = x² + 1, du = 2x dx, then ∫u³du = u⁴/4 + C
Frequently Asked Questions
What is Integration by Substitution Calculator?
- Integration by Substitution Calculator evaluates Integration by Substitution for the math / integration library. Primary input cue: Function f(x) [required]: Function to integrate. Method anchor: Substitution method: Let u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du After integration, substitute back u = g(x). Substitution simplifies integration .
How does Integration by Substitution Calculator work?
- Substitution method: Let u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du After integration, substitute back u = g(x). Substitution simplifies integration by changing variable (calculator id: integration-by-substitution-calculator).
Why use this Math Numbers calculator?
- Use Integration by Substitution Calculator when you already know the inputs and need integration by substitution without setting up a spreadsheet.