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(x^(m))/("log"(e)x)...

`(x^(m))/("log"_(e)x)`

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(cos x)/("log"_(e)x)

Find the range of f(x)=log_(e)x-((log_(e)x)^(2))/(|log_(e)x|)

Knowledge Check

  • Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(1)x^(m)(log_(e).(1)/(x))dx=

    A
    `(f(n+1))/((m+1)^(n))`
    B
    `(f(n))/((m+1)^(n+1))`
    C
    `(f(n+1))/((m+1)^(n+1))`
    D
    `g(m+1),n+1)`

  • The integral int_(1)^(e){((x)/(e))^(2x)-((e)/(x))^(x)} "log"_(e)x dx is equal to

    A
    `(3)/(2)-e-(1)/(2e^(2))`
    B
    `-(1)/(2)+(1)/(e)-(1)/(2e^(2))`
    C
    `(1)/(2)-e-(1)/(e^(2))`
    D
    `(3)/(2)-(1)/(e)-(1)/(2e^(2))`

  • If f(x)=log_(e)(log_(e)x)/log_(e)x then f'(x) at x = e is

    A
    `0`
    B
    `1`
    C
    `e`
    D
    `1//e`

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    Find the range of f(x)=(log)_(e)x-(((log)_(e)x)^(2))/(|(log)_(e)x|)

    int((1+(log)_(e)x)^(2))/(1+(log)_(e)x^(x+1)+((log)_(e)x^(sqrt(x)))^(2))dx=

    I=int(log_(e)(log_(e)x))/(x(log_(e)x))dx

    Evaluate: int(e^(5)(log)_(e)x-e^(4)(log)_(e)x)/(e^(3)(log)_(e)x-e^(2)(log)_(e^(x))x)dx

    Evaluate: int(e^(6)(log)_(e)x-e^(5)(log)_(e)x)/(e^(4)(log)_(e)x-e^(3)(log)_(e^(x))x)dx

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