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Prove that int0^(pi//2)sin^2x/(sinx+cosx...

Prove that `int_0^(pi//2)sin^2x/(sinx+cosx)dx=int_0^(pi//2)(cos^2x)/(sinx+cosx)dx`.

Answer

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Knowledge Check

  • I_(1(n))=int_(0)^(pi//2)(sin(2n-1)x)/(sinx)dx,I_(2(n))=int_(0)^(pi//2)(sin^(2)nx)/(sin^(2)x)dx,n in N then

    A
    `I_(2(n+1))-I_(2(n))=I_(1(n))`
    B
    `I_(2(n+1))-I_(2(n))=I_(1(n+1))`
    C
    `I_(2(n+1))+I_(1(n))=I_(2(n))`
    D
    `I_(2(n+1))+I_(1(n+1))=I_(2(n))`

  • int_(-pi//2)^(pi//2)sin^(2)xcos^(2)x(sinx+cosx)dx=

    A
    `(2)/(15)`
    B
    `4/15`
    C
    `2/5`
    D
    `(8)/(15)`

  • int_(0)^(1)(tan^(-1)x)/(x)dx is equal to a) int_(0)^(pi/2)(sinx)/(x)dx b) int_(0)^(pi/2)(x)/(sinx)dx c) 1/2int_(0)^(pi/2)(sinx)/(x)dx d) 1/2int_(0)^(pi/2)(x)/(sinx)dx

    A
    `int_(0)^(pi/2)(sinx)/(x)dx`
    B
    `int_(0)^(pi/2)(x)/(sinx)dx`
    C
    `1/2int_(0)^(pi/2)(sinx)/(x)dx`
    D
    `1/2int_(0)^(pi/2)(x)/(sinx)dx`

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