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int(0)^([x]//3) (8^(x))/(2^([3x]))dx whe...

`int_(0)^([x]//3) (8^(x))/(2^([3x]))dx` where [.] denotes the greatest integer function, is equal to

A

`([x])/("In"2)`

B

`([x])/("In"2)`

C

`(2[x])/("In"2)`

D

`([x])/("In"8)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=underset(0)overset([x]//3)int(8^(x))/(2^([3x]))dx`.Then,
`I=underset(0)overset([x]//3)int 2^(3x-[3x])dx`
`rArr I=[x]underset(0)overset(1//3)int 2^(3x-[3x])dx [:' 2^(3x-[3x])" is periodic with period "1//3]`
`rArr I=[x]underset(0)overset(1//3)int 2^(3x)dx=[x][(2^(3x))/(3 log 2)]_(0)^(1//3)=([x])/(log 8)`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
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  2. If k in N and I(k)=int(-2kp)^(2kpi) |sin x|[sin x]dx, where [.] denote...

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  3. The value of int(-1)^(1) ((logx+sqrt(1+x^(2))))/(x+log(x+sqrt(1+x^(2...

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  4. If int(0)^(1) alphae^(betax^(2))sin(x+k)dx=0 for some alpha,beta in R,...

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  5. int(0)^([x]//3) (8^(x))/(2^([3x]))dx where [.] denotes the greatest in...

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  6. Let f(x)=In[cos|x|+(1)/(2)] where [.] denotes the greatest integer fun...

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  7. lim(x to 0)(int(0^(x) x e^(t^(2))dt)/(1+x-e^(x)) is equal to

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  8. If int(2x^(2))^(x^(3)) (In x)f(t) dt=x^(2)-2x+5, then f(8)=

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  9. lim(x to 0)(int(-x)^(x) f(t)dt)/(int(0)^(2x) f(t+4)dt) is equal to

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  10. If f(x+f(y))=f(x)+y for all x,y in R and f(0)=1, then int(0)^(10) f(10...

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  11. If alpha,beta(beta gt alpha) are the roots of f(x)=-ax^(2)+bx+c=0 and ...

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  12. The value of the constant a gt 0 such that int(0)^(a) [tan^(-1)sqrt(x)...

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  13. If f(x) is a continuous function in [0,pi] such that f(0)=f(x)=0, the...

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  14. Let f:R to R be continuous function such that f(x)=f(2x) for all x in ...

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  15. The value of int(0)^(pi//4) (tan^(n)x+tan^(n-2)x)d(x-([x])/(1!)+([x]...

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  16. The value of the definite integral int(t+2pi)^(t+5pi//2) {sin^(-1)(c...

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  17. If f(x) is an integrable function on [(pi)/(6),(pi)/(3)] and I(1)=in...

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  18. Let f(x)=lim(n to oo ) {(n^(n)(x+n)(x+(n)/(2))....(x+(n)/(2)))/(n!(...

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  19. The total number of distinct x in [0,1] for which int(0)^(x) (t^(2))...

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  20. For x in R,x ne 0, if y(x) is differentiable function such that xint(1...

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