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Find the value of int(0)^(4)[x]dx, where...

Find the value of `int_(0)^(4)[x]dx`, where `[.]` represents the gretest integer function.

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Verified by Experts

The correct Answer is:
6
Graph of `y=[x]` for `0ltxlt4` is as shown in the following figure.

From the graph `int_(0)^(4)[x]dx=0` (for `0ltxlt1)`
`+(1xx1)` (for `1ltxlt2`)
`+(1xx2)` (for `2ltxlt3`)
`+(1xx3)` (for `3ltxlt4`)
`=0+1+2+3`
`=6`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
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  2. Evaluate the following integrals . int(a)^(b)x^(3)dx

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  3. Find the value of int(0)^(4)[x]dx, where [.] represents the gretest in...

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  4. If f(x)={1-|x|, |x|lt=1 and 0 , |x|>1 and g(x)=f(x-1)+f(x+1), find ...

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  5. Consider the integral int0^(2pi)(dx)/(5-2cosx) making the substitution...

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  6. Evaluate the following : int(0)^(pi)(dx)/(1+sinx)

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  7. Evaluate: int1^oo (e^(x+1)+e^(3-x))^(-1)dx

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  8. Evaluate: int0^(1/(sqrt(2)))(sin^(-1)x)/((1-x^2)sqrt(1-x^2))dx

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  9. Evaluate: int0^1(2-x^2)/((1+x)sqrt(1-x^2))dx

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  10. Evaluate the following : int(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2...

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  11. Evaluate: int(pi//6)^(pi//4)(1+cotx)/(e^(x)sinx) dx

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  12. Evaluate int(0)^(1)(e^(-x)dx)/(1+e^(x))

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  13. Prove that int0^(102)(x-1)(x-2)..(x-100)xx(1/(x-1)+1/(x-2)+.... .+ 1/...

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  14. Show that : int0^1(logx)/((1+x))dx=-int0^1(log(1+x))/x dx

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  15. If int0^1(e^t)/(1+t)dt=a , then find the value of int0^1(e^t)/((1+t)^2...

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  16. Let f be a one-to-one continuous function such that f(2)=3 and f(5)=7....

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  17. Evaluate: ("lim")(n rarr oo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))+ .......

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  18. Evaluate: underset(nrarrinfty)lim[(1/n^2sec^2(1/(n^2))+2/n^2sec^2((4)/...

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  19. Evaluate ("lim")(n rarr oo)sum(k=1)^nk/(n^2+k^2)

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  20. Evaluate the following limit: lim(nto oo)(sum(r=1)^(n) sqrt(r)sum(r=...

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