Evaluate the following definite integrals 2 x 1 cos

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Unformatted text preview: ( x ) dx = 2 sin( x )dx = − 2 2 cos x + C ∫ ∫2 2 2 2 2 [5 pts each]. Evaluate the following definite integrals. π /2 ∫ ( x + 1) cos xdx (a) −π / 2 π /2 π /2 −π / 2 − /2 ∫ x cos xdx + = ∫ cos xdx = 0 + sin x | π π π /2 − /2 = 1 − (−1) = 2 Note that: y= x cos x is an odd function. π /2 ∫ sin 2 x cos xdx (b) 0 = π /2 ∫ 2 sin x cos x cos xdx Let t = cosx, dt = - sinx dx 0 0 10 = −2∫ t 2dt = −2 t 3 | = 0 − (−2 / 3) = 2 / 3 31 1 1 ∫ x tan (c) −1 xdx 0 1 1 1 1 π 1 x2 π1 1 = ∫ tan −1 xd ( x 2 ) = x 2 tan −1 x | − ∫ x 2 d (tan −1 x) = − ∫ dx = − ∫ 1 − dx 2 2 2 2 8 2 0 1+ x 8 2 0 1 + x2 0 0 0 1 = 1 π 1 1 1 −1 1 π 1 π π 1 − x | + tan x | = − + = − 8 202 82842 0 1 (d) ∫ 0 x3 x2 +1 dx 1 1 Let t = x + 1 ⇒ dt = 2 xdx ⇒ dx =...
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This note was uploaded on 01/22/2014 for the course MTH 153 taught by Professor Wu during the Fall '10 term at Michigan State University.

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