 # 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)$\int \frac{3{e}^{x}}{{e}^{x}-1}dx$
ii) ${\int }_{0}^{1}X\mathrm{cos}\left(\pi X\right)dX$
iii)$\underset{x\to 1}{lim}\frac{x-1}{\sqrt{x}-1}$
iv)$\underset{x\to \mathrm{\infty }}{lim}\frac{x}{\mathrm{ln}\left(1+3{e}^{x}\right)}$
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i)$=3\int \frac{1}{u}du=3\mathrm{ln}\left(u\right)+c=3\mathrm{ln}\left({e}^{x}-1\right)+c$
$u=\mathrm{exp}\left(x\right)-1$
$du=\mathrm{exp}\left(x\right)dx$
ii) $=x\ast -\int \frac{\mathrm{sin}\left(\pi x\right)}{\pi }dx=x+{\frac{\mathrm{cos}\left(\pi x\right)}{{\pi }^{2}}}_{0}^{1}=\frac{-2}{{\pi }^{2}}$
$\mathrm{cos}\left(\pi x\right)dx=dv$
dx=du
$\frac{\mathrm{sin}\left(\pi x\right)}{\pi }=v$
iii) = 0/0 L'Hopital
$\underset{x\to 1}{lim}\frac{1}{\frac{1}{2\sqrt{x}}}=\underset{x\to 1}{lim}2\sqrt{x}=2$
$\underset{x\to \mathrm{\infty }}{lim}\frac{x}{\mathrm{ln}\left(1+3{e}^{x}\right)}=\mathrm{\infty }/\mathrm{\infty }{L}^{\prime }Hopital$
iv)$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{\frac{3{e}^{x}}{3{e}^{x}+1}}=\underset{x\to \mathrm{\infty }}{lim}\frac{3{e}^{x}+1}{3{e}^{x}}=\mathrm{\infty }/\mathrm{\infty }{L}^{\prime }Hopital$
$\underset{x\to \mathrm{\infty }}{lim}\frac{3{e}^{x}}{3{e}^{x}}=3$