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Prove that : tan^(-1)((sqrt(1+x^(2))+s...

Prove that :
`tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))=pi/4+1/2cos^(-1)x^(2)`.

Answer

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Prove that : tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x)))=pi/4-1/2cos^(-1)x,-1/sqrt2lexle1

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Knowledge Check

  • int(1+x^(2))/(sqrt(1-x^(2)))dx=...

    A
    `(3)/(2)sin^(-1)x-(1)/(2)x sqrt(1-x^(2))+c`
    B
    `(3)/(2)sin^(-1)x+(1)/(2)x sqrt(1-x^(2))+c`
    C
    `(3)/(2)cos^(-1)e-(1)/(2)x sqrt(1-x^(2))+c`
    D
    `(3)/(2)cos^(-1)x+(1)/(2)x sqrt(1-x^(2))+c`

  • int(x^(3))/(sqrt(1+x^(2)))dx=a(1+x^(2))^((3)/(2))+b* sqrt(1+x^(2))+C

    A
    `a=(1)/(3) and b=1`
    B
    `a=-(1)/(3) and b=1`
    C
    `a=-(1)/(3) and b=-1`
    D
    `a=(1)/(3) and b=-1`

  • int(x)/(sqrt(1+x^(2)+sqrt((1+x^(2))^(3)) )) dx=....+c

    A
    `(1)/(2) log|+sqrt(1+x^(2))|`
    B
    `2 sqrt(1+sqrt(1+x^(2)))`
    C
    `2(1+sqrt(1+x^(2)))`
    D
    None of these

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