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

Prove that `int_(a)^(b)(x)dx = int_(a)^(b)f(a+b-x)dx` and `int_(pi/4)^(pi/3)(dx)/(1+sqrt(tanx))`.

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int_(pi//6)^(pi//3) (dx)/(1 + sqrt(tan x)) =

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`

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

    A
    `pi/3`
    B
    `pi/6`
    C
    `pi/12`
    D
    `pi/2`

  • int(a^(-x)-b^(-x))dx =

    A
    `(a^(-x))/log a-(b^(-x))/log b`
    B
    `(a^(-x)-b^(-x)) (log a - log b)`
    C
    `(b^(-x))/log b - (a^(-x))/log a`
    D
    `b^(-x)-a^(-x)`

    • Similar Questions

    Explore conceptually related problems

    Prove that int_(a)^(b) f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx

    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.

    Prove that int_(a)^(b) f(x)dx= int_(a)^(b) f (a+b-x)dx" hence evaluate " int_(0)^(pi/4) log(1+tan x)dx .

    Prove that int_(0)^(a)(x)dx = int_(0)^(a) f(a-x)dx and hence evaluate int_(0)^(pi/4)log (1 + tan x)dx .

    Prove that int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx and hence evaluate int_(0)^(pi//2)(2log sin x-log sin2x)dx .

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