int(1)/(x)(log(ex)e)dx is equal to...

`int(1)/(x)(log_(ex)e)dx` is equal to

A

`log_(e)(1-log_(e)x)+C`

B

`log_(e)(log_(e)ex-1)+C`

C

`log_(e)(log_(e)x-1)+C`

D

`log_(e)(log_(e)x+1)+C`

Text Solution

Verified by Experts

The correct Answer is:

D

Let `l=int(1)/(x)(log_(ex)e)dx=int(1)/(x(1+log_(e)x))dx`
Put `log_(e)x=t" "rArr" "(1)/(x)dx=dt`
`therefore" "l=int(dt)/((1+t))=log_(e)(1+t)+C`
`=log_(e)(1+log_(e)x)+C`

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The value of int _(1)^(e) 10^(log_(e)x) dx is equal to

A

`10 log_(e) (10e)`

B

`(10e-1)/(log_(e)10e)`

C

`(10e)/(log_(e)10 e)`

D

`(10 e) log_(e) (10e)`

int{sin(log_(e)x)+cos(log_(e)x)}dx is equal to

A

`sin(log_(e)x)+cos(log_(e)x)+C`

B

`xsin (log_(e)x)+C`

C

`xcos(log_(e)x)+C`

D

none of these

The integral int_(1)^(e){((x)/(e))^(2x)-((e)/(x))^(x)} "log"_(e)x dx is equal to

A

`(3)/(2)-e-(1)/(2e^(2))`

B

`-(1)/(2)+(1)/(e)-(1)/(2e^(2))`

C

`(1)/(2)-e-(1)/(e^(2))`

D

`(3)/(2)-(1)/(e)-(1)/(2e^(2))`

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