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"Let " f(x)=int x^(sinx)(1+xcosx*In x+si...

`"Let " f(x)=int x^(sinx)(1+xcosx*In x+sinx)dx " and " f((pi)/(2))=(pi^(2))/(4). " 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 f(x)=(1-cosx)/(1-sinx)," then: "f'((pi)/(2)) is

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    indeterminate

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    `1-1/sqrt2(tan^(-1))1/sqrt2`
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    `1+1/sqrt2(tan^(-1))1/sqrt2`

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