Integration by u Substitution Definite Integral Calculator
Master Integration by u Substitution Definite Integrals
Integration by u substitution is a powerful technique used to evaluate definite integrals. It’s particularly useful when the integrand contains a function of a function, or when the integral can be simplified by making a substitution.
How to Use This Calculator
- Enter the function u and its derivative du.
- Enter the lower and upper limits of integration.
- Click the Calculate button.
Formula & Methodology
The formula for integration by u substitution is:
∫[f(u) * du] = ∫[g(v)] dv
where u = g(v) and du = g'(v) dv.
Real-World Examples
Example 1: Evaluate ∫(3x2 + 2x) dx from 0 to 2.
Let u = x3 + x2. Then du = (3x2 + 2x) dx.
∫(3x2 + 2x) dx = ∫du = u = x3 + x2 from 0 to 2.
Example 2: Evaluate ∫(x2 + 1) / (x3 + 1) dx from 1 to 2.
Let u = x3 + 1. Then du = 3x2 dx.
∫(x2 + 1) / (x3 + 1) dx = ∫(1 / u) du = ln|u| from 1 to 2.
Example 3: Evaluate ∫(sin(x)) / (cos(x)) dx from 0 to π/4.
Let u = cos(x). Then du = -sin(x) dx.
∫(sin(x)) / (cos(x)) dx = -∫(1 / u) du = -ln|u| from 1 to cos(π/4).
Data & Statistics
| Technique | Ease of Use | Accuracy | Speed |
|---|---|---|---|
| Integration by u substitution | 3 | 5 | 4 |
| Integration by parts | 4 | 4 | 3 |
| Numerical integration | 2 | 3 | 5 |
Expert Tips
- Always check if the integral can be evaluated using other techniques before attempting integration by u substitution.
- Be careful when choosing the substitution. It should simplify the integral and make it easier to evaluate.
- After making the substitution, check if the new integral is easier to evaluate.
Interactive FAQ
What is integration by u substitution?
Integration by u substitution is a technique used to evaluate definite integrals by making a substitution that simplifies the integral.
When should I use integration by u substitution?
You should use integration by u substitution when the integrand contains a function of a function, or when the integral can be simplified by making a substitution.
Learn more about integration by u substitution
Watch a video tutorial on integration by u substitution