integral of (e to the power 5x into 5 lnx +1/x)dx
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Now we integrate by parts:
\begin{aligned} &\phantom{=}\displaystyle\int_0^5 xe^{-x}\,dx \\\\ &=\displaystyle\int_0^5 u\,dv \\\\ &=\Big[uv\Big]_0^5-\displaystyle\int_0^5 v\,du \\\\ &=\displaystyle\Big[ -xe^{-x}\Big]_0^5-\int_0^5-e^{-x}\,dx \\\\ &=\Big[-xe^{-x}-e^{-x}\Big]_0^5 \\\\ &=\Big[-e^{-x}(x+1)\Big]_0^5 \\\\ &=-e^{-5}(6)+e^0(1) \\\\ &=-6e^{-5}+1 \end{aligned}
=∫
0
5
xe
−x
dx
=∫
0
5
udv
=[uv]
0
5
−∫
0
5
vdu
=[−xe
−x
]
0
5
−∫
0
5
−e
−x
dx
=[−xe
−x
−e
−x
]
0
5
=[−e
−x
(x+1)]
0
5
=−e
−5
(6)+e
0
(1)
=−6e
−5
+1
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