MATH1051 Lecture 6: Lecture 06
5.5 – Substitution
Review
●What is the chain rule?
○Take the derivative of (x) sin(x)f= 2
○(x) cos(x) xf′= 2*2
●Undoing the chain rule
○Is harder to recognize when to undo the chain rule when finding the integral
●Instead, you can rewrite certain parts
○ = os(x)2xdx
∫
c2os(u)du
∫
c
○How?
■Set xu = 2
■Then take derivative of u
■x
dx
du = 2
■Then isolate du
■u 2xdxd =
■And substitute
○os(u)du
∫
c
■Finish taking integral
■in(u) Cs +
■Replace u’s
■in(x) Cs 2+
Examples
●an (x)sec (x)dx
∫
t7 2
○ tanxu =
○u sec dxd = 2
○du
∫
u7
○u) C(8
18+
○(tanx) C
8
18+
●dx
∫
1+x2
tan x
−1
