Integration by Substitution
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1 November 22, 2013
2 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =?
3 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4
4 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4 f (x) = 1 7 (x 2 + 5) 4
5 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4 f (x) = 1 7 (x 2 + 5) 4 f (x) = ln(x 2 + 1)
6 Anti-differentiation formula An anti-differentiation formula is of the form f (x)dx = F (x) + C
7 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4
8 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4 f (x) = e x2 +4 2x
9 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4 f (x) = e x2 +4 2x f (x)dx = f (x) + C.
10 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = e x2 +4 f (x) = e x2 +4 2x f (x)dx = f (x) + C. e x xdx = e x C.
11 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = 1 7 (x 2 + 5) 4
12 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = 1 7 (x 2 + 5) (x 2 + 5) 3 2xdx = 1 7 (x 2 + 5) 4 + C.
13 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = ln(x 2 + 1)
14 Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation formula. f (x) = ln(x 2 + 1) 1 x xdx = ln(x 2 + 1) + C.
15 Notice The derivative of each inside function is 2x in the example. Notice that the derivative of the inside function is a factor in the integrand in each anti-differentiation formula.
16 Notice The derivative of each inside function is 2x in the example. Notice that the derivative of the inside function is a factor in the integrand in each anti-differentiation formula. Finding an inside function whose derivative appears as a factor in the integrand is key to the method of the substitution.
17 Make a substitution in an integral Let w (or v, t,...) be the inside function.
18 Make a substitution in an integral Let w (or v, t,...) be the inside function. dw = w (x)dx or dx = 1 w (x) dw
19 Make a substitution in an integral Let w (or v, t,...) be the inside function. dw = w (x)dx or dx = 1 w (x) dw Express the integrand in terms of w.
20 Examples e x 2 2xdx =? (x 2 + 6) 3 2xdx =? 1 2xdx =? x 2 +6
21 Example In the previous examples the derivative of the inside function must be present in the integrand for this method to work. Our method works even when the derivative is missing a constant factor. 4te t2 +3 dt =?
22 Example x 3 3 x 4 + 7dx =?
23 Example x 2 dx =? 1 + x 3
24 Examples sin(3 x)dx =? e 2 4x dx =?
25 Examples 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =?
26 Example e cos θ sin θdθ =?
27 Examples cos x x dx =? e x x dx =?
28 Examples e x e x dx =? + 1 e x + 1 e x dx =? + x
29 Examples sin n (kx) cos(kx)dx =? cos n (kx) sin(kx)dx =?
30 Harder Examples 1 dx =? 1 + ex 1 dx =? 2 + 3x x ln xdx =?
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