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Let f:[0,oo)vecR be a continuous strictl...

Let `f:[0,oo)vecR` be a continuous strictly increasing function, such that `f^3(x)=int_0^x tdotf^2(t)dt` for every `xgeq0.` Then value of `f(6)` is_______

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Verified by Experts

The correct Answer is:
6
Given `f^(3)(x)=int_(0)^(x)=int_(0)^(x)t.f^(2)(t)dt`
Differentiating `3f^(2)(x)f'(x)=xf^(2)(x)`
`f(x)!=0`
`:.f'(x)=x/3:.f(x)=(x^(2))/6+3`
But `f(0)=0impliesC=0`
`f(6)=6`
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CENGAGE-DEFINITE INTEGRATION -Exercise (Numerical)
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  3. Let f:[0,oo)vecR be a continuous strictly increasing function, such th...

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  4. If f is continuous function and F(x)=int0^x((2t+3)dotintt^2f(u)d u)dt ...

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  5. If the value of the definite integral int0^1(sin^(-1)sqrt(x))/(x^2-x+1...

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  6. about to only mathematics

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  7. Let g(x) be differentiable on R and int(sint)^1x^2g(x)dx=(1-sint), wh...

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  8. If int0^oox^(2n+1)dote^(-xdx)=360 , then the value of n is

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  9. Let f(x) be a derivable function satisfying f(x)=int0^x e^tsin(x-t)dt...

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  10. Let f(x)=1/x^2 int0^x (4t^2-2f'(t))dt then find f'(4)

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  11. If the value of the definite integral int0^1^(2007)C7x^(2000)dot(1-x)^...

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  12. IfIn=int0^1(1-x^5)^n dx ,t h e n(55)/7(I(10))/(I(11))i se q u a lto

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  13. Evaluate: 5050(int0 1(1-x^(50))^(100)dx)/(int0 1(1-x^(50))^(101)dx)

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  14. L e tJ=int(-5)^(-4)(3-x^2)tan(3-x^2)dxa n dK=int(-2)^(-1)(6-6x+x^2) t...

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  15. The value of the definite integral int(2-1)^(sqrt(2)+1)(x^4+x^2+2)/((x...

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  16. Consider a real valued continuous function f such that f(x)=sinx + int...

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  17. Iff(x)=x+int0^1t(x+t)f(t)dt ,t h e nt h ev a l u eof(23)/2f(0) is ...

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  18. Let y=f(x)=4x^(3)+2x-6, then the value of int(0)^(2)f(x)dx+int(0)^(30)...

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  19. The value of int(1)^(3)(sqrt(1+(x-1)^(3))+(x^(2)-1)^(1/3)+1)dx is .

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  20. The value of int(0)^(1)xcos^(-1)((1-x^2)/(1+x^2)) dx (x>0) is equal to...

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