Question

integral of (e to the power 5x into 5 lnx +1/x)dx​

Answer 1

Answer:

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