How Do You Integrate by Substitution?

• French B
• Geography
• German A
• German A Lit
• German ab initio
• German B
• Global Politics
• History
• History (First Exam 2028)
• Mathematics Analysis and Approaches (AA)
• Mathematics Applications & Interpretation (AI)

• Philosophy
• Physics
• Psychology
• Psychology (First Exam 2027)
• Spanish A
• Spanish A Lit
• Spanish ab initio
• Spanish B
• Sports, exercise and health science (SEHS)
• Theory of Knowledge (TOK)
• World Religions

• MYP English Lang & Lit
• MYP Extended Mathematics
• MYP History

• MYP Individuals & Societies
• MYP Integrated Sciences
• MYP Physics

• MYP Standard Mathematics

This works because an integrand containing both $g(x)$ and its derivative (g'(x)) can often be rewritten as a simpler function of $u$.

A good substitution makes the remaining expression depend only on $u$. If an $x$-term remains, either rewrite it using the substitution or choose a different $u$. Constant factors may be moved outside the integral.

Use these steps:

1. Choose $u$ to simplify the integrand, usually taking the inner function of a composite expression.
2. Differentiate the substitution to find (du).
3. Rewrite every occurrence of $x$, including (dx), in terms of $u$.
4. Integrate with respect to $u$.
5. For an indefinite integral, substitute back to $x$ and include $+C$.

For a definite integral, either change the limits into $u$-values or substitute back before applying the original limits. Do not mix these approaches.

Worked Example

Evaluate

Let $u=x+1$. Therefore, $x=u-1$ and $du=dx$. The integral becomes

Integrating term by term gives

Substituting $u=x+1$ back:

Differentiate the result to check it: applying the chain rule should recover the original integrand. This is a quick way to detect a missing factor, incorrect power, or incomplete substitution.

A common misconception is that choosing $u$ alone completes the substitution. It does not: the entire integrand and differential must be converted into $u$.

IB Exam Technique

This is IB Mathematics AA HL, AHL 5.16, and may appear on any HL paper. For the command term Integrate, show the substitution, the transformed integral, and the final antiderivative; on Paper 1, keep the work exact, and never omit $+C$ for an indefinite integral.

At AHL, a substitution may be provided; use it explicitly. For definite integrals, show the transformed limits clearly.