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Substitution

Substitution is the chain rule used backwards. If you see a function and its derivative in the same integral, you set the inner function equal to $u$ . The integral then becomes a power or an exponential function in $u$ . In a definite integral you can put $x$ back before you use the limits. You check by differentiating the answer: the inner derivative should come back.

\int g'(x)\, f(g(x))\,dx = \int f(u)\,du

substitution

Symbols

$u$	the inner function
$g'(x),dx$	du

Example

$\int_0^1 2x(x^2 + 1)^3\,dx$ : with $u = x^2 + 1$ the answer is $\left[\dfrac{(x^2 + 1)^4}{4}\right]_0^1 = \dfrac{16 - 1}{4} = 3.75$ .

The factor in front of the inner function must be its derivative, otherwise a constant is missing in the substitution.

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Part of Mathematics R2: Integration techniques.