6.2Integration bySubstitution & Separable Differential EquationsM.L.King Jr. Birthplace, Atlanta, GAGreg KellyHanford High SchoolRichland, WashingtonPhoto by Vickie Kelly, 2002
The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate..
Example 1:(2952xdx+∫Let 2ux=+dudx=5u du∫616uC+(29626xC++The variable of integration must match the variable in the expression.Don’tforget to substitute the value for uback into the problem!→
Example:(Exploration 1 in the book)212 xx dx+⋅∫One of the clues that we look for is if we can find a function and its derivative in the integrand.The derivative of is .21x+2 x dx12udu∫3223uC+(29322213xC++2Let 1ux=+2 dux dx=Note that this only worked because of the 2xin the original.