U Substitution Calculator

The U Substitution Calculator applies the substitution method for integrals. It is useful when an integrand contains a function together with its derivative, allowing a complicated expression to be rewritten in a simpler variable. Use the tool to work through the calculation and review the result.

Definite
Indefinite

with respect to

Upper value

Lower value

Basic rule

Let u = g(x). Then du = g'(x) dx, and an integral involving g(x) and g'(x) can often be rewritten as an integral in u.

Worked example

Consider ∫ 2x(x² + 1)³ dx. Set u = x² + 1, so du = 2x dx. The integral becomes ∫u³ du = u⁴/4 + C. Substituting back gives (x² + 1)⁴/4 + C.

When substitution works well

Look for an inner expression whose derivative also appears, possibly multiplied by a constant. For definite integrals, the limits can be changed to the new variable so a separate back-substitution is unnecessary.

Frequently Asked Questions FAQ

When is U substitution useful?

It works well when an inner expression appears together with its derivative, allowing the integral to be rewritten in a simpler variable.

What is a worked U-substitution example?

For ∫2x(x²+1)³dx, let u=x²+1. Then du=2x dx and the integral becomes ∫u³du=(u⁴/4)+C.

How do I choose u?

A useful choice is often the inner function whose derivative is also present in the integrand, possibly multiplied by a constant.

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