5.22 Integration by substitution

The basic idea is that we can replace the independent variable $x$ by $u=u(x)$ and obtain

When writing an integration by substitution:

(a) state what the substitution is;

(b) differentiate $u$ to obtain $du/dx$;

(c) change variables, all in one go;

(d) change limits for definite integrals; or

(e) convert back to the original variable for an indefinite integral, simplifying the functions where possible.

Solution.