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Let f(x)=intx^(sinx)(1+xcosxdotlnx+sinx)...

Let `f(x)=intx^(sinx)(1+xcosxdotlnx+sinx)dxa n df(pi/2)=(pi^2)/4dot` Then the value of `|"cos"(f(pi))|` is____

Text Solution

Verified by Experts

The correct Answer is:
-1
`f(x)=int x^(sinx)(1+xcosx*Inx+sinx)dx`
`"If " F(x)=x^(sinx)=e^(sinx Inx)," then " `
`f(x)=int(F(x)+xF'(x))=xF(x)+C`
`=x*x^(sinx)+C`
`"or " f((pi)/(2))=(pi)/(2)*(pi)/(2)+C " or " C=0`
` :. f(x)=x(x)^(sinx),f(pi)=pi(pi)^(0)=pi`
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Knowledge Check

  • If int_(0)^(pi) x f(sinx) dx=A int_(0)^((pi)/(2))f(sinx)dx , then the value of A is -

    A
    `(pi)/(4)`
    B
    `(pi)/(2)`
    C
    `pi`
    D
    `2pi`

  • If f(x)=cos[pi^2]x+cos[-pi^2]x then the value of f(pi/4)+f(pi/2) is

    A
    `1/sqrt2`
    B
    `1+1/sqrt2`
    C
    `1-1/sqrt2`
    D
    `-1+1/sqrt2`

  • Let f(x)=sin^(4)x-cos^(4)x int_(0)^((pi)/(2))f(x)dx =

    A
    `(3pi)/(16)`
    B
    `(3pi)/(8)`
    C
    0
    D
    `(pi)/(16)`

    • Similar Questions

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    If f(x)=|x|^(|sinx|), then find f^(prime)(-pi/4)

    Ifint_(sinx)^1t^2f(t)dt=1-sinx ,w h e r e .x in (0,pi/2), then find the value of f(1/(sqrt(3)))dot

    Ifint_(sinx)^1t^2f(t)dt=1-sinx ,w h e r e .x in (0,pi/2), then find the value of f(1/(sqrt(3)))dot

    The value of the integral sqrt2 int_0^(pi/2) f(sin2x) sinx dx=A (sqrt2/9) int_0^(pi/4) f(cos2x)cosx dx then the value of A is

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