5.6 Notes - Substitution for Definite Integrals (Section...

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Unformatted text preview: Substitution for Definite Integrals (Section 5.6) Theorem 6: If g′ is continuous on the interval [ a, b ] and f is continuous on the range b of g , then ∫ f ( g( x ))g′( x ) dx = a g (b ) ∫ f (u ) du . g(a ) Using Theorem 6: 4 Example 1: Evaluate ∫ 3x + 4 dx . 0 Step 1: Identify g( x ) and set u = g( x ) as before. Step 2: Find du. Step 3: Evaluate g(a ) and g(b ) . b g (b ) a g(a ) Step 4: Use theorem 6 to rewrite ∫ f ( g( x ))g′( x ) dx as ∫ f (u ) du . Step 5: Evaluate the integral. **Note that for DEFINITE integrals, you do not substitute back in at the end. π 4 Example 2: Evaluate ∫ 0 tan 3 θ dθ . cos 2 θ 4 Example 3: Evaluate ∫ x 5 − x dx . 1 Example 4: Evaluate the following: eπ / 4 4 a) ∫ dt t (1 + ln 2 t ) 1 b) c) π 2 π − 2 cos y dy sin y + 2 ∫ ∫ 3 0 1 dx x + 2x + 1 2 Completing the square Rewrite x 2 + bx + c as ( x + b )2 + c − ( b )2 . 2 2 Example 5: Complete the square for each of the following: a) x 2 + x + 3 b) 3 − x 2 + x 3 Example 6: Evaluate: ∫ 2 dx 3x + x + 3 ...
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