Integration by Substitution Calculator

Solve common integrals using u-substitution and see every transformation step.

Calculate an Integral

Enter the integrand and the expression you want to use for u.

Examples: 2x*sin(x^2), 3/(3x+1), 2x*(x^2+1)^5
Usually choose the inner function.

Try These Examples

How u-Substitution Works

Integration by substitution, also called u-substitution, is a change-of-variable technique used to simplify integrals.

If u = g(x) then du = g′(x) dx .

The goal is to replace the matching derivative factor in the original integral with du, leaving a simpler integral in terms of u.

Example: ∫ 2x sin(x²) dx

Choose: u = x²
Therefore: du = 2x dx

The integral becomes: ∫ sin(u) du

Therefore: −cos(u) + C

Finally: −cos(x²) + C

Common u-Substitution Patterns

Integral Useful substitution
2x sin(x²) u = x²
2x e^(x²) u = x²
2x/(x²+1) u = x²+1
3/(3x+1) u = 3x+1
3 cos(3x+1) u = 3x+1
cos(x) sin(x) u = sin(x)
2x(x²+1)^5 u = x²+1

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Frequently Asked Questions

U-substitution is a change-of-variable technique that replaces a useful inner expression with u and its differential with du.

Look for an inner expression whose derivative also appears somewhere in the integrand. For example, for 2x sin(x²), choose u = x² because du = 2x dx.

No. This calculator focuses specifically on common educational u-substitution patterns. It is not intended to replace a complete computer algebra system.