Evaluate the following: i) int (3e^x)/(e^x -1)dx ii) int_0^1 Xcos(pi X)dX iii) lim_(xrightarrow 1)(x-1)/(sqrt(x)-1) iv) lim_(xrightarrow infty) x/(ln (1+3e^x))

Nathalie Fields 2022-07-25 Answered
Evaluate the following:
i) 3 e x e x 1 d x
ii) 0 1 X cos ( π X ) d X
iii) lim x 1 x 1 x 1
iv) lim x x ln ( 1 + 3 e x )
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Answers (1)

edgarovhg
Answered 2022-07-26 Author has 12 answers
i) = 3 1 u d u = 3 ln ( u ) + c = 3 ln ( e x 1 ) + c
u = exp ( x ) 1
d u = exp ( x ) d x
ii) = x sin ( π x ) π d x = x + cos ( π x ) π 2 0 1 = 2 π 2
cos ( π x ) d x = d v
dx=du
sin ( π x ) π = v
iii) = 0/0 L'Hopital
lim x 1 1 1 2 x = lim x 1 2 x = 2
lim x x ln ( 1 + 3 e x ) = / L H o p i t a l
iv) lim x 1 3 e x 3 e x + 1 = lim x 3 e x + 1 3 e x = / L H o p i t a l
lim x 3 e x 3 e x = 3
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