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If =int(0)^(1) x^(n)e^(-x)dx "for" n in ...

If =`int_(0)^(1) x^(n)e^(-x)dx "for" n in N "then" I_(n)-nI_(n-1)=`

A

e

B

`1//e`

C

`-1//e`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
We have
`I_(n)=underset(0)overset(1)int underset(I)x^(n)underset(II)e^(-x)dx=[-x^(n)e^(-x)]_(0)^(1)+n underset(0)overset(1)intx^(-1)e^(-x)dx`
`rArr I_(n)=-e^(-1)+nI_(n-1)rArr I_(n)-nI_(n)=-1//e`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
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  2. If =int(1)^(e) (logx)^(n) dx, "then"I(n)+nI(n-1) is equal to

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  20. int(-1)^(10) sgn (x-[x])dx equals

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