Integration by u Substitution Definite Integral Calculator

u: du: Limits:

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

Comparison of Integration Techniques
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