The U Substitution Calculator assists with integrals that contain an inner expression together with its derivative. U-substitution is essentially the integration counterpart of the chain rule and is often the quickest way to simplify a composite integrand.
The Basic Idea
Choose an expression u that makes the integral simpler. Then compute du and rewrite the integral using u. The target form is something like ∫g(u)du, which can be integrated using familiar rules before substituting the original expression back.
Example
Consider ∫2x(x²+1)³dx. Choose u=x²+1, so du=2x dx. The integral becomes ∫u³du = u⁴/4 + C. Substituting back gives (x²+1)⁴/4 + C.
How to Use the Calculator
Enter the integral and any substitution information requested by the tool. Check that the chosen u-expression and its differential match the original integrand. A substitution that leaves extra complicated terms may not be the best choice.
Connection With the Chain Rule
If differentiating a composite function multiplies by the derivative of the inner expression, integration can reverse that pattern when the inner derivative is present. Recognizing this structure is often more valuable than memorizing a list of isolated substitutions.
Verify the Answer
Differentiate the returned antiderivative. If the derivative returns the original integrand, including its constant factors and signs, the integration step is consistent.