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If =int(1)^(e) (logx)^(n) dx, "then"I(n...

If =`int_(1)^(e) (logx)^(n) dx, "then"I_(n)+nI_(n-1)` is equal to

A

`1//e`

B

e

C

`e-1`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We have
`=underset(1)overset(e)int underset(I)((log x))^(n)underset(II)1 dx`
`rArr I_(n)=[x(logx)^(n)]_(1)^(e)-underset(1)overset(e)int n (logx)^(n-1)*(1)/(x)x dx`
`rArr I_(n)=e-n underset(1)overset(e)int (logx)^(n-1)dx=e-nI_(n)-1`
`rArr I_(n)+nI_(n-1)=e`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. The value of int(-1)^(1) (d)/(dx) ("tan"^(1)(1)/(x))dx is

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  2. The vlaue of the integral int(-1)^(3) ("tan"^(1)(x)/(x^(2)+1)+"tan"^...

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  3. If =int(1)^(e) (logx)^(n) dx, "then"I(n)+nI(n-1) is equal to

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  4. If =int(0)^(1) x^(n)e^(-x)dx "for" n in N "then" I(n)-nI(n-1)=

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  5. The value of int(1//n)^((an-1)//n) (sqrt(x))/(sqrt(a-x+sqrtx))dx, is

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  6. The value of the integral int(0)^(pi//2)log |tan x cot x |dx is

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  7. If I(1)=int(x)^(1)(1)/(1+t^(2)) dt and I(2)=int(1)^(1//x) dt "for" x g...

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  8. For all values of , int(1//e)^(tanx) (t)/(1+t^(2))dt+int(1//e)^(tanx) ...

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  9. The absolute value of int(10)^(19) (cosx)/(1+x^(8))dx, is

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  10. If f(x) is an odd pefiodc function defined on the interval [T/2,T/2], ...

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  11. If int(pi//2)^(theta) sin x dx=sin 2 theta then the of theta satisfyin...

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  12. If f(x) is periodic function with period, T, then

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  13. If f(n)=(1)/(n){(n+1)(n+2)(n+3)...(n+n)}^(1//n) then lim(n to oo)f(n)...

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  14. The poins of extremum of phi(x)=int(1)^(x) e^(-t^(2)//2) (1-t^(2))dt a...

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  15. int(-2)^(2) min(x-[x],-x-[x])dx equals, where [x] represents greates i...

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  16. The integral int(0)^(a) (g(x))/(f(x)+f(a-x))dx vanishes, if

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  17. If (1)/(sqrt(a))int(1)^(a)((3)/(2)sqrt(x)+1-(1)/(sqrt(x)))dx lt4 then ...

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  18. The expression (int(0)^(n)[x]dx)/(int(0)^(n){x}dx)a where [x] and [x] ...

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  19. If f(x)= Min {|x-1|,|x|,|x+1|} "then" int(-1)^(1)f(x)dx eqals

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  20. The value of int(0)^(100) [ tan^(-1)x]dx is

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