Calculator Inputs
Use common functions like sin(u), cos(u), exp(u), log(u), sqrt(u), and powers such as u^2 or x^3.
Plotly Graph
The graph displays the transformed integrand against the substitution variable u.
Example Data Table
| Original Problem | Substitution | Transformed Integral | x-Bounds | u-Bounds | Approximate Value |
|---|---|---|---|---|---|
| ∫₀² (x²+1)² · 2x dx | u = x² + 1 | ∫₁⁵ u² du | [0, 2] | [1, 5] | 41.3333333333 |
| ∫₀^(π/2) sin(x)cos(x) dx | u = sin(x) | ∫₀¹ u du | [0, π/2] | [0, 1] | 0.5 |
| ∫₁³ 1/x dx | u = x | ∫₁³ 1/u du | [1, 3] | [1, 3] | 1.0986122887 |
Formula Used
The change of variables rule rewrites an integral by substituting a new variable. If u = g(x), then du = g'(x) dx.
For indefinite integrals:
∫ f(g(x))g'(x) dx = ∫ f(u) du
For definite integrals:
∫ from x=a to x=b of f(g(x))g'(x) dx = ∫ from u=g(a) to u=g(b) of f(u) du
This calculator numerically evaluates the transformed definite integral with Simpson’s Rule after converting bounds from x to u.
How to Use This Calculator
- Enter the substitution function u(x), such as x^2+1 or sin(x).
- Enter du/dx for reference, so the displayed steps match your working.
- Type the transformed integrand in u, such as u^2 or 1/u.
- Choose definite mode to evaluate a numeric answer.
- Enter the original x-bounds. Use automatic or manual u-bounds.
- Press Compute Integral to show the result above the form.
- Review the graph, exported CSV, or PDF summary if needed.
FAQs
1. What does a change of variables integral calculator do?
It helps rewrite an integral using a substitution variable, converts bounds when needed, and evaluates the transformed integral numerically. It is especially useful for checking algebra, comparing bounds, and visualizing the integrand after substitution.
2. When should I use substitution in integration?
Use substitution when part of the integrand is a function and another part resembles its derivative. Common patterns include compositions like (x²+1)^n·2x, sin(x)cos(x), or exponential and logarithmic combinations that simplify after replacing an inner expression.
3. Why do the bounds change in a definite integral?
Once the variable changes from x to u, the limits must also match the new variable. You evaluate u at the original x-lower and x-upper values. That keeps the definite integral consistent and avoids converting back after integration.
4. Can this tool determine the net change between the given values of the variable calculator problem?
Yes. In definite mode, the calculator returns the integral value over the transformed interval. That value represents the net change across the chosen bounds, provided your transformed integrand correctly models the rate or accumulation quantity.
5. What functions can I enter in the transformed integrand?
You can use powers, parentheses, and functions such as sin, cos, tan, sqrt, exp, log, abs, and log10. Enter the expression in terms of u only for the transformed integral field, and use x only in the substitution field.
6. Is the displayed answer exact or approximate?
The definite integral is evaluated numerically with Simpson’s Rule, so it is an approximation. Increasing the interval count usually improves accuracy. The setup and bound conversion remain symbolic in appearance, but the final numeric value is computed approximately.
7. What if my substitution is correct but the result looks wrong?
Check the transformed integrand carefully. Many errors happen when dx is not fully replaced, a derivative factor is missing, or bounds are not converted correctly. Also verify whether you selected automatic or manual u-bounds appropriately.
8. Can I use this for learning as well as homework checking?
Yes. It is useful for practice, verification, and step review. You can compare original and transformed intervals, inspect the graph, and export results. Still, always write the substitution steps yourself so you understand the reasoning behind the method.