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The smallest interval [a,b] such that ...

The smallest interval [a,b] such that
`overset(1)underset(0)int (1)/(sqrt(1+x^(4)))dx in [a,b]`, is

A

`[1//sqrt(2),1]`

B

[0,1]

C

`[1//2,1]`

D

[3//4,1]`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I=overset(1)underset(0)int (1)/(sqrt(1+x^(4)))dx`.
We have,
`0 le x le 1 rArr0 le x^(4) le 1 rArr 1 le 1+x^(4) le 2`
`rArr1 le sqrt(1+x^(4))le sqrt(2)`
`rArr(1)/(sqrt(2))le(1)/(sqrt(1+x^(4))) le 1`
`(1)/(sqrt(2))(1-0) le overset(1)underset(0)int (1)/(sqrt(1+x^(4)))dx le 1(1-0)`
`(1)/(sqrt(2)) le 1 le 1 rArr I in [(1)/(sqrt(2)),1]`
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OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Chapter Test 2
  1. The smallest interval [a,b] such that overset(1)underset(0)int (1)/...

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  2. The value of the integral int(0)^(2)x[x]dx

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  3. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  4. Let f (x) be a function satisfying f(x)=f(x) with f(0) = 1 and g be th...

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  5. If I=int(0)^(1)cos(2 cot^(-1)sqrt(((1-x)/(1+x))))dx then :

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  6. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

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  7. The vaue of int(-1)^(2) (|x|)/(x)dx is

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  8. The value of int0^1 (x^(3))/(1+x^(8))dx is

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  9. The value of int(0)^(3) xsqrt(1+x)dx, is

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  10. Evaluate int(0)^(1)log(sin((pix)/(2)))dx

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  11. Evaluate int(0)^(pi) xlog sinx dx

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  12. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)=underset(0)overset(oo)i...

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  13. If f(x)={{:(x,xlt1),(x-1,xge1):}, then underset(0)overset(2)intx^(2)f(...

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  14. The value of the integral overset(1)underset(0)int (1)/((1+x^(2))^(3//...

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  15. Prove that: int0^(2a)f(x)dx=int0^(2a)f(2a-x)dxdot

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  16. If int(0)^(36) (1)/(2x+9)dx =log k, is equal to

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  17. The value of the integral int(0)^(pi//2) sin^(6) x dx, is

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  18. If int(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int(0)^(oo) e^(-ax^(2)...

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  19. The value of the integral int 0^oo 1/(1+x^4)dx is

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  20. The value of alpha in [0,2pi] which does not satify the equation int(p...

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  21. lim(x to 0)(int(0)^(x^(2))sinsqrt(t) dt)/(x^(3)) is equl to

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