Unformatted text preview: that the numerator has degree less than 2. Now we can complete the square to turn the integral into a trigonometric substitution problem. EXAMPLE 8.20 Evaluate Z x x 2 + x + 1 dx . We have seen that this quadratic does not factor. We complete the square: x 2 + x + 1 = x 2 + x + 1 4 + 11 4 = x 2 + x + 1 4 + 3 4 = ± x + 1 2 ¶ 2 + 3 4 . Now factor out 3 / 4: 3 4 ˆ 4 3 ± x + 1 2 ¶ 2 + 1 ! = 3 4 ˆ ± 2 √ 3 x + 1 √ 3 ¶ 2 + 1 ! . Now let tan u = 2 √ 3 x + 1 √ 3 sec 2 u du = 2 √ 3 dx dx = √ 3 2 sec 2 u du....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Division, Fractions, Rational Functions, Fraction, Rational function, dx, ln x

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