Integral U Sub Calculator

This integral u-sub calculator helps you solve integrals using substitution (u-substitution) with step-by-step solutions. Learn how to apply the u-sub method, understand the formula, and visualize your results with interactive charts.

How to Use This Calculator

To use the integral u-sub calculator:

Enter the integrand in the input field (e.g., x² + 3x + 2)
Select the substitution variable (u)
Enter the substitution expression (e.g., u = x² + 3x + 2)
Click "Calculate" to see the step-by-step solution
Review the result and chart visualization

The calculator will show you the complete u-substitution process, including:

The substitution step (du = ...dx)
The rewritten integral in terms of u
The final antiderivative
The evaluated result

The u-sub Formula

The u-substitution method is based on the chain rule in reverse. The general formula is:

∫f(x) dx = ∫f(g(u)) g'(u) du

Where:

u = g(x)
du = g'(x) dx

To use u-substitution:

Choose a substitution u = g(x) that simplifies the integrand
Find du by differentiating u with respect to x
Rewrite the integral in terms of u
Integrate with respect to u
Substitute back for x

Worked Examples

Example 1: Basic u-substitution

Find ∫(2x + 1)³ dx

Let u = (2x + 1)³
Then du = 3(2x + 1)²(2) dx = 6(2x + 1)² dx
So dx = du / [6(2x + 1)²]
But we have (2x + 1)³ dx, so multiply by (2x + 1)³ / (2x + 1)³ = 1
Thus, ∫(2x + 1)³ dx = ∫u du / [6(2x + 1)²] → Wait, this seems incorrect. Let's correct:
Actually, we have ∫(2x + 1)³ dx = ∫u du / [6(2x + 1)²] → This is not correct. The proper approach is:
Let u = 2x + 1 → du = 2 dx → dx = du/2
Then ∫(2x + 1)³ dx = ∫u³ (du/2) = (1/2)∫u³ du = (1/2)(u⁴/4) + C = u⁴/8 + C
Substitute back: (2x + 1)⁴/8 + C

Example 2: More complex substitution

Find ∫x²√(x³ + 4) dx

Let u = x³ + 4 → du = 3x² dx → x² dx = du/3
Thus, ∫x²√(x³ + 4) dx = ∫√u (du/3) = (1/3)∫u^(1/2) du
Integrate: (1/3)(2/3)u^(3/2) + C = (2/9)u^(3/2) + C
Substitute back: (2/9)(x³ + 4)^(3/2) + C

Frequently Asked Questions

What is u-substitution in calculus?: U-substitution is a technique in integral calculus that uses substitution to simplify complex integrals. It's based on the chain rule in reverse and allows you to rewrite an integral in terms of a simpler variable.
When should I use u-substitution?: Use u-substitution when the integrand contains a composite function (a function inside another function) that, when substituted, simplifies the integral. Look for patterns like ∫f(g(x))g'(x) dx.
What if my integral doesn't fit the u-substitution pattern?: If your integral doesn't clearly fit the u-substitution pattern, try other integration techniques like integration by parts, trigonometric substitutions, or partial fractions. Sometimes, a substitution might not be immediately obvious.
How do I know if I've chosen the right substitution?: The best substitution is one that simplifies the integrand. Look for parts of the integrand that can be set equal to u, and whose derivative appears elsewhere in the integrand. Practice helps develop intuition for choosing good substitutions.
Can u-substitution be used for definite integrals?: Yes, u-substitution works for both definite and indefinite integrals. When working with definite integrals, remember to change the limits of integration according to the substitution you've chosen.