How Do You Integrate By Inspection (Reverse Chain Rule)?

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so reversing it gives

This works because differentiating the final expression reproduces the original integrand.

In IB Mathematics: Analysis and Approaches, this is SL 5.10 and therefore applies at both SL and HL. It may be tested analytically on Paper 1 or within a longer problem on Paper 2.

Recognition Strategy

Use these steps:

1. Find a likely inner function $g(x)$, usually inside a power, exponential, logarithm, or trigonometric function.
2. Differentiate it to obtain (g'(x)).
3. Compare (g'(x)) with the remaining factor. If it differs only by a constant, factor that constant outside.
4. Integrate the outer pattern and verify by differentiation.

Worked Examples

For

choose $g(x)=x^2+1$, so $g'(x)=2x$. Therefore,

For

the inner derivative is $2x$, while $4x=2(2x)$. Hence,

Constant Adjustment Example

A useful exponential pattern is

Take $g(x)=3x+1$, so $g'(x)=3$. The integrand is missing this factor, so compensate by multiplying outside by $\frac{1}{3}$. This gives $\frac{1}{3}e^{3x+1}+C$. The adjustment is valid because the mismatch is a constant; differentiating the answer returns $e^{3x+1}$. This is the standard constant-adjustment method expected in analytic IB working.

When Inspection Does Not Work

Inspection only works when the required derivative factor is present up to a constant multiple. For example, $\int (x^2+1)^4\,dx$ does not match because the required factor $2x$ is absent. You cannot insert a variable factor and compensate with a constant. By contrast, $\int x(x^2+1)^4\,dx$ matches, giving $\frac{(x^2+1)^5}{10}+C$. Always compare factors before treating an expression as a composite.

Common Patterns

IB Exam Technique

A common misconception is ignoring the inner derivative. Show the inner function or constant adjustment before integrating. Differentiate your result mentally, include $+C$ for every indefinite integral, and remember that technology does not replace the algebraic setup.