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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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 Fall '08
 Justin
 Fractions, Derivative, ln x, Henning Hohnhold, UCSD Henning Hohnhold

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