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09 - Substitution - u du"" cos x 2 2 x dx = cos u...

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Substitution Suppose we want to find 2 t cos t 2 ( ) dt ! . The basic rules only tell us how to find cos t ( ) dt ! . We need to introduce a new variable so that we can use the basic integration formulas. Let u = t 2 and du = 2t dt. Then 2 t cos t 2 ( ) dt ! = cos u du = sin u + C = sin t 2 ( ) + C ! Substitution Rule: If u = g(x) is a differentiable function whose range is on an interval I and f is continuous on I, then f ( g ( x )) ! g x ( ) dx = f ( u ) du
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Unformatted text preview: u ) du " " cos x 2 ( ) 2 x ( ) ! dx = cos u du ! ( ) Ex. 1 e x ! dx = Ex. t 2 1 ! 5 t 3 3 " dt = Ex. t 1 ! 5 t 3 3 " dt = Ex. sin x cos x ! dx = Ex. 1 2 ! 3 s " ds = Ex. e x tan e x ( ) ! ln cos e x ( ) dx = Do: 1. Ex. e t 1 + e t ! dt = 2. tan ! 1 x 1 + x 2 " dx = 3. 1 + 4 x 1 + 2 x + 4 x 2 ! dx = 4. cos 2 2 y + 3 ( ) ! sin 2 y + 3 ( ) dy = 5. cot z ! dz =...
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