CAS MA 123 Lecture 38: Substitution for Indefinite Integrals

• School: Boston University
• Department: Mathematics & Statistics
• Course: CAS MA 123
• Professor: John Bergdall

Document Summary

Ma123 lecture 38 substitution for indefinite integrals. Suppose f(x) is an antiderirative for f(x) find an antiderivative of g(x)=f(3x) G(x) = 1/3 f(3x) is an antiderivative for g(x) = f(3x) Why? d/dx (g(x)) = d/dx (1/3 f(3x)) = 1/3 d/dx (f(3x) = 1/3 * 3 * f"(3x)= f"(3x) = f(3x)= g(x) D dx ( f ( x)) dx= f "( x)dx =f ( x)+c. F ( x) dx=f ( x )+c >>> f ( g( x))g" ( x) dx=f(g ( x))+c. Sin2 xcosx dx g(x) = sinx f(x) = x2. F ( g( x)) g"( x) dx so substitution says ( x) sin . Where f(x) is an antiderivative of f(x) = x2 x3+c >>> sin2 xcosx dx= 1. 2 xe x2 dx >>> f(x2) + c where composition f(g(x)) f(x) = ex g(x) = x2 g"(x) = 2x. Dx in f ( x ) dx x in reimann sum.