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13.2 Integration by Substitution • Let w be the inside function • Write dw w' ( x) dx • Do cross-multiply • Back to the given integral and do the substitution • Take the integral • Express the final answer in term of x ( 2 x 3 ) ( 6 x ) dx ? 3 20 2 (2 x w 3 dw 6x 2 dx dw 6 x dx 2 3) (6 x )dx 20 20 dw 2 5 e dx ? 5x 5 e dx 5x dw 5 dx e dw dw 5dx e C w w e C 5x 1 4 x 2 2 xdx ? dw 2x dx dw 2 xdx 1 4 x 2 2 xdx 1 dw w ln w C ln 4 x C 2 ( x 3 x 7 ) ( 2 x 3 ) dx ? 2 4 ( x 3 x 7) w dw 2 dw 2x 3 dx dw (2 x 3)dx 4 ( 2 x 3)dx 4 5 w C 5 ( x 3 x 7) C 5 2 5 ( 2 x 3 ) ( 6 x ) dx ? 3 4 2 ( 2 x 3 ) ( 6 x ) dx 3 4 2 w dw 4 dw 2 6x dx dw 6 x dx 2 5 w C 5 (2 x 3) C 5 3 5 e dw 3 dx dw 3dx dw dx 3 1 dw dx 3 3 x dx ? 3 x e dx 1 e dw 1 e w dw 3 3 w 1 w e C 3 1 3 x e C 3 x dx 2 x 9 dw 2x dx dw 2 xdx dw xdx 2 1 dw xdx 2 x x 2 9dx 1 1 dw w 2 1 1 dw 2 w 1 ln w C 2 1 ln x 2 9 C 2 0.001e 0.01x dx ? .01x . 001 e dx dw .01 dx dw .01dx dw dx .01 100dw dx .001e w (100dw) 0.1e dw w 0.1e C w 0.1e .01x C • Yah! We are done. • What have you studied in MAT 116? • What is Calculus? • What is “calculus”? – Calculus is the study of how things change. • What do you learn in a calculus class? – Functions, limits, derivatives, integrals, … • Derivatives can used to study applications that involve the rate of change such as population growth, radioactive decay, optimization problems, … • Integrals can be used to calculate the area, equation of a curve, business sales, … • How do algebra and calculus differ? Ex: Solving equations to find a particular value of a variable Ex: How a change in a variable affects another variable?