University of Illinois at Urbana Champaign Fall

University of Illinois at Urbana-Champaign Fall 2006 Math 380 Group G1 Graded Homework V Correction. 1. Compute the derivative of the function x 7? tan?1(x) = arctan(x) ; use it to compute ∫ b a dx x2 + 1 , where a, b ? R (in terms of arctan(a), arctan(b)), then to compute ∫ 1 0 dx x2 + x + 1 . With a change of variable, compute the integral ∫ pi 2 0 cos(x)dx 2? cos2(x) + sin(x) . Correction. A direct computation shows that tan?(x) = 1 + tan2(x). The fact that tan(arctan(x)) = x, and the Chain Rule for functions of one real variable, yields tan?(arctan(x)). arctan?(x) = 1, so that arctan?(x) = 1 tan?(arctan(x)) = 1 1 + (tan(arctan(x)))2 = 1 1 + x2 . This immediately yields ∫ b a dx x2 + 1 = arctan(b)? arctan(a). One has ∫ 1 0 dx x2 + x + 1 = ∫ 1 0 dx (x + 12 ) 2 + 34 = ∫ 3 2 1 2 dy y2 + 34

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