Integration by substitution is the integration equivalent of the chain rule in differentiation. It is one of the most powerful tools in your A-Level Maths toolkit, designed to transform a complicated integral that you cannot solve directly into a standard form that is simple to calculate.
- Core Rule: Let $u = g(x)$ for the composite part of the function (usually inside brackets, roots, or denominators).
- Differential: Differentiate $u$ to find $\frac{du}{dx}$, rearrange to substitute for $dx$.
- Definite Integrals: Always change the limits of integration from $x$-limits to $u$-limits immediately.
- Boards: Key topic for OCR A/B, AQA, and Edexcel Paper 1 & 2.
The Core Concept: When & Why to Use Substitution
We use substitution when the integrand contains a function and its derivative (or a multiple of it) multiplied together. Formally, we look for integrals in the form:
By defining $u = g(x)$, we find the differential relationship $\frac{du}{dx} = g'(x)$, which simplifies the integral into $\int f(u) \, du$.
The 4-Step Method
Choose your substitution $u$
Look for the "inner function" — the part of the expression enclosed in a bracket, power, square root, or denominator. Set $u$ equal to this function.
Differentiate $u$ to replace $dx$
Find $\frac{du}{dx}$. Rearrange this expression to express $dx$ in terms of $du$ and $x$ (e.g. $dx = \frac{du}{g'(x)}$).
Substitute everything and simplify
Replace $g(x)$ with $u$, and $dx$ with your expression from Step 2. Simplify the terms. All $x$ terms should cancel out, leaving you with an integral containing only $u$.
Change limits (Definite) or back-substitute (Indefinite)
For definite integrals, use $u = g(x)$ to recalculate your upper and lower limits. For indefinite integrals, integrate with respect to $u$, then substitute $g(x)$ back in for $u$ at the end, adding the constant of integration $+ C$.
Worked Exam Example: Definite Integral
Let's evaluate the following typical A-level exam question:
Step 1: Choose $u$
The inner function inside the bracket is $x^2 + 1$. We set:
Step 2: Differentiate to find $dx$
Differentiating $u$ with respect to $x$ gives:
Step 3: Convert the integration limits
This is a definite integral, so we must convert the $x$-limits ($x = 0$ and $x = 2$) into $u$-limits using $u = x^2 + 1$:
- Upper limit: when $x = 2$, $u = (2)^2 + 1 = 5$
- Lower limit: when $x = 0$, $u = (0)^2 + 1 = 1$
Step 4: Substitute and integrate
Now rewrite the integral with our new components:
Notice that the $x$ terms cancel out nicely:
Evaluate the limits:
Common Pitfall: Leaving limits unchanged
Many students integrate with respect to $u$ but evaluate using the original $x$ limits (0 and 2 instead of 1 and 5). This leads to incorrect answers. Always calculate your new limits immediately in Step 3!
Pro-Tip: Look for the derivative first
Before writing anything down, check if the derivative of the inner function is outside. In our example, the derivative of $x^2+1$ is $2x$. The expression has an $x$ outside. Since $x$ is a multiple of $2x$, substitution is guaranteed to work cleanly.
Frequently Asked Questions
If $x$ terms remain, you need to express those remaining $x$ terms in terms of $u$ by rearranging your initial substitution equation. For example, if $u = x + 2$, then $x = u - 2$. Replace the remaining $x$'s with $(u - 2)$ and expand.
No. Constant of integration ($+ C$) is only required for indefinite integrals (integrals without upper and lower boundary numbers).
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