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Fall 2007 - Hohnhold's Class - Quiz 4 (Version A)

Fall 2007 - Hohnhold's Class - Quiz 4 (Version A) - Letting...

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Quiz 4 for Math 20B Fall Quarter 2007, UCSD Henning Hohnhold Name: Section: #1: #2: Total: (1) Use partial fractions to evaluate the integral Z 4 - 1 x x 2 - 2 x - 15 dx. The partial fractions decomposition takes the form x x 2 - 2 x - 15 = x ( x - 5)( x + 3) = A x - 5 + B x + 3 . From this, we get x = A ( x + 3) + B ( x - 5) and choosing x = 5 and x = - 3 we obtain A = 5 8 and B = 3 8 . Now we can integrate an find Z 4 - 1 x x 2 - 2 x - 15 dx = 1 8 Z 4 - 1 5 x - 5 + 3 x + 3 dx = 1 8 (5 ln | x - 5 | + 3 ln | x + 3 | ) | 4 - 1 = 1 8 (3 ln(7) - 5 ln(6) - 3 ln(2))
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(2) Evaluate the integral Z 4 (1 + x 2 ) 2 dx by using the tangent substitution x = tan θ and simplifying the integrand using
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Unformatted text preview: Letting x = tan( θ ) we have 1 + x 2 = 1 + tan 2 ( θ ) = sec 2 ( θ ) and dx = sec 2 ( θ ) dθ so that Z 4 (1 + x 2 ) 2 dx = Z 4 sec 4 ( θ ) sec 2 ( θ ) dθ = Z 4 sec 2 ( θ ) dθ = Z 4 cos 2 ( θ ) dθ = 2 Z 1 + cos(2 θ ) dθ = 2( θ + 1 2 sin(2 θ )) + C = 2 arctan( x ) + sin(2 arctan( x )) + C = 2 arctan( x ) + 2 x 1 + x 2 + C (The last simplification is not required.) 2 Good Luck! 3...
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