Answer to Question #139643 in Calculus for Timmy

Question

An integration method that essentially involves using the chain rule in reverse is called…..

Answer 1

$\displaystyle\textsf{Integration by Algebraic Substitution}\\ \textrm{Explanation:}\\ \textsf{For an Integral}\, S\\ S = \int f(x) \, \mathrm{d}x\\ \textsf{make the algebraic substitution}\\x = g(u)\\ g'(u) \mathrm{d}u = \mathrm{d}x\\ \therefore S = \int f(g(u)) g'(u)\mathrm{d}u \\ \textbf{\textsf{Example}}\\ \textsf{Integrate}\, \cos(r^2) r \mathrm{d}r\\ I = \int \cos(r^2) r \mathrm{d}r\\ \textsf{Let}\, u = r^2, \mathrm{d}u = 2r \mathrm{d}r \\ \begin{aligned} I &= \int \frac{\cos(u)}{2r} r \mathrm{d}u \\&= \int \frac{\cos(u)}{2}du \\&= \frac{\sin(u)}{2} + C \\&= \frac{\sin(r^2)}{2} + C \end{aligned}$