Substitution, colloquially referred to as “u-sub”,
is a technique of integration useful when the integrand contains a composite of functions.
Essentially it helps us infer the antiderivative of a function
by “undoing” the chain rule of differentiation:
\(\displaystyle \int {f'\bigl({\color{SteelBlue}{g(x)}}\bigr)\color{SteelBlue}{g'(x)}} \,\mathrm{d}x = f\bigl({\color{SteelBlue}{g(x)}}\bigr) + C \)
Sometimes a substitution is referred to as a change of coordinates,
in reference to the geometric perspective of this algebraic operation.
An Example
\(\displaystyle \int x^4\sqrt{x^5+7} \,\mathrm{d}x
\;=\; \int \sqrt{{\color{SteelBlue}{x^5+7}}} \,\,{\color{SteelBlue}{x^4}} \,\mathrm{d}x
\;=\; \int \sqrt{{\color{SteelBlue}{u}}} \,\, {\color{SteelBlue}{\tfrac{1}{5}\frac{\mathrm{d}u}{\cancel{\mathrm{d}x}}}} \,\cancel{\mathrm{d}x}
\;=\; {\color{SteelBlue}{\tfrac{1}{5}}} \int \sqrt{{\color{SteelBlue}{u}}} \,{\color{SteelBlue}{\mathrm{d}u}}
\;=\; {\color{SteelBlue}{\tfrac{1}{5}}} \tfrac{2}{3} {\color{SteelBlue}{u}}^{3/2} + C
\;=\; \tfrac{2}{15} \bigl(x^5+7\bigr)^{3/2} + C \)
Letting \(u = x^5+7,\) for which \(\frac{\mathrm{d}u}{\mathrm{d}x} = 5x^4,\)
because of the extra factor of \(5\) we use the substitution \(\tfrac{1}{5}\frac{\mathrm{d}u}{\mathrm{d}x} = x^4.\)
For a definite integral, the bounds must change in accordance with the substitution.
If the previous example were instead a definite integral \(\int_0^2,\)
those \(x\)-bounds of \(0\) and \(2\) would have to change to the \(u\)-bounds \(\int_7^{39}\)
per the substitution rule \(u = x^5+7.\)