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Answer 1

4.05

Explanation:

(a)

`\boxed{(d)/(dx) (u\cdot v)=u\cdot dv+v\cdot du}`

`(d)/(dx) (x\cdot ln\ x)=x((1)/(x) )+1(ln\ x)`

`=1+ln\ x`

(b)

since
`(d)/(dx) (x\cdot ln\ x)=1+ln\ x`

therefore
`\int {(d)/(dx) (x\cdot ln\ x)=\int {(1+ln\ x)} \, dx }`

`\int {(d)/(dx) (x\cdot ln\ x)=\int {1} \, dx } +\int {ln\ x} \, dx }`

`x\cdot ln\ x=x+\int {ln\ x} \, dx }`

`\int {ln\ x} \, dx } =x\cdot ln\ x-x`

`\int\limits^5_1 {ln\ x} \, dx =(5(ln\ 5)-5)-(1(ln\ 1)-1)`

`=5(ln\ 5)-ln\ 1 -5+1`

`=5(ln\ 5)-ln\ 1 -4`

`=4.05`