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

# Fall 2007 - Hohnhold's Class - Quiz 4 (Version B) - 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 1 - 5 x x 2 + 3 x - 18 dx. The partial fractions decomposition takes the form x x 2 + 3 x - 18 = x ( x + 6)( x - 3) = A x + 6 + B x - 3 . From this, we get x = A ( x - 3) + B ( x + 6) and choosing x = - 6 and x = 3 we obtain A = 2 3 and B = 1 3 . Now we can integrate an ﬁnd Z 1 - 5 x x 2 + 3 x - 18 dx = 1 3 Z 1 - 5 2 x + 6 + 1 x - 3 dx = 1 3 (2 ln | x + 6 | + ln | x - 3 | ) | 1 - 5 = 1 3 (2 ln(7) + ln(2) - ln(8))

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(2) Evaluate the integral Z 6 (1 + x 2 ) 2 dx by using the tangent substitution x = tan θ and simplifying the integrand using the deﬁnition of sec.
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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 6 (1 + x 2 ) 2 dx = Z 6 sec 4 ( θ ) sec 2 ( θ ) dθ = Z 6 sec 2 ( θ ) dθ = Z 6 cos 2 ( θ ) dθ = 3 Z 1 + cos(2 θ ) dθ = 3( θ + 1 2 sin(2 θ )) + C = 3 arctan( x ) + 3 2 sin(2 arctan( x )) + C = 3 arctan( x ) + 3 x 1 + x 2 + C (The last simpliﬁcation is not required.) 2 Good Luck! 3...
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Fall 2007 - Hohnhold's Class - Quiz 4 (Version B) - Letting...

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