Unformatted text preview: Substitution for Indefinite Integrals (Section 5.5) () 2
Suppose we want to find ∫ 2t cos t 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 = t2 and du = 2t dt. Then ∫ 2t cos (t ) dt = ∫ cos u du = sin u +C = sin (t ) + C 2 2 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 ( ∫ cos ( x )(2 x ) dx = ∫ cos u du ) 2 Example 1: Find ∫ sec x tan x
dx . sec x Step 1: Identify g( x ) and set u = g( x ). Ways to identify u: 1) Parentheses 2) Square root sign 3) Denominators 4) Exponents 5) Log and trig. Functions Step 2: Find du. Step 3: Rewrite integral in terms of u and du. Step 4: Integrate in terms of u. Step 5: Rewrite answer in terms of the original variable. Remarks: 1) The original variable must be the variable in the final solution. (i.e. make sure to do step 5 above) 2) Your final answer is still an antiderivative, so you can check your solution by differentiating. Example 2: Integrate the following: a) ∫ t 2 (t 3 + 7 )15 dt b) ∫ sin c) ∫ t2 + 1 d) ∫ ln x
dx
x e) ∫ 5 x cos x dx 2t dt x2 dx
1− x Example 3: (Substitution can’t do everything!) Evaluate ∫ sin 2 x dx . Example 4: Evaluate the following: 1⎞
⎛
a) ∫ ⎜ (−4 z − 5 )3 / 2 +
⎟ dz
⎝
7z + 4 ⎠ b) ∫ 7 e5 − 2 r dr ⎛
⎞
x
c) ∫ ⎜ 3 / 2
2
⎟ dx
⎝ ( x + 1) + 1 ⎠
4 d) ∫ (e x2 − 4 x + 4 ) ( x − 2 ) dx ⎛z⎞
e) ∫ ⎜ 2
dz
⎝ z − 4⎟
⎠ ...
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 '06
 EDeSturler
 Integrals, Formulas, Trigraph, Natural logarithm, dx, 2t cos

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