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Prove that int(a)^(b)f(x)dx=int(a)^(b)f(...

Prove that `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx` and hence evaluate `int_((pi)/(6))^((pi)/(3))(1)/(1+sqrt(tanx))dx.`

Answer

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Evaluate : int_((pi)/(6))^((pi)/(3))(sinx+cosx)/(sqrt(sin2x))dx.

a) Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx" and evaluate "int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx)) b) Prove that |{:(1+a^(2)-b^(2), 2ab, -2b), (2ab, 1-a^(2)+b^(2), 2a), (2, -2a, 1-a^(2)-b^(2)):}|=(1+a^(2)+b^(2))^(3)

Knowledge Check

  • int_(pi/6)^(pi/3) (dx)/(1+sqrttanx) =

    A
    `pi/12`
    B
    `pi/2`
    C
    `pi/6`
    D
    `pi/4`

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