This section is about the Substitution rule which allows anti-differentiation of complex expressions. The idea is to replace the complex section with a variable, (u), anti-differentiate, and then substitute back in the complex statement: a sort of inverse chain rule.
Substitution Rule for indefinite integrals
If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I then
∫ f(g(x))g'(x) dx = ∫ f(u) du
This rule can also be applied to definite integrals by adjusting the range for u
∫ from a to be of [f(g(x))g'(x) dx] = ∫ from g(a) to g(b) of [f(u) duExample
∫ (cos√t)/ √t
Given that u = √t then dt = 2du/t^-.5
substituting both expressions in removes the denominator and the roots to give "∫ 2cos(u) du"
antidifferentiate to get 2sin(u) + C and substitute the original expression for x to get 2sin√t + C
This is a helpful site: http://www.sosmath.com/calculus/integration/substitution/substitution.html
Alex you're next.