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int(0)^(pi//2)sinx sin2x sin3x sin 4x dx...

`int_(0)^(pi//2)sinx sin2x sin3x sin 4x dx=`

A

`(pi)/(4)`

B

`(pi)/(8)`

C

`(pi)/(16)`

D

`(pi)/(32)`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int_(0)^(pi//2)sinx sin2x sin3x sin 4x dx=int_(0)^(pi//2)cos x sin 2x cos3x sin 4x dx`
`rArr" "2I=int_(0)^(pi//2)cos2xsin2xsin4xdx`
`=(1)/(2)int_(0)^(pi//2)sin^(2)4xdx`
`=(1)/(2)int_(0)^(pi//2)(1-cos8x)/(2)dx`
`=(1)/(4)[x-(sin8x)/(8)]_(0)^(pi//2)=(pi)/(8)`
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int_0^(pi/2) sin x sin 2x dx

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Knowledge Check

  • int_(0)^(pi//2)sin x.sin 2x dx=

    A
    `(4)/(3)`
    B
    `(1)/(3)`
    C
    `(3)/(4)`
    D
    `(2)/(3)`

  • int_(0)^(pi//2) (sinx )/(sin x + cos x ) dx=

    A
    `pi/4`
    B
    `pi/2`
    C
    `(3pi)/4`
    D
    `(3pi)/2`

  • int_(pi//3)^(pi//2) dx/(sinx+ sin 2x)=

    A
    `2/3 log 2- 1/2 log 3`
    B
    `4/3 log 2 - 1/2 log 3 `
    C
    `2/3 log 2 - 3/2 log 3 `
    D
    `4/3log 2- 3/2log 3`

    • Similar Questions

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    int_(0)^( pi/4)sin2x*sin3xdx

    " int sinx sin 2x (sin3x)dx

    int_(0)^( pi/2)((sin x cos x) /(1+sin 4x ))dx

    Evaluate the following : int_(0)^(pi//2)(sinx cos x)/((2sinx+1)(sin x+1))dx .

    int_(0)^(pi//2) f(sin 2x) sin x dx= int_(0)^(pi//2) f(sin 2x) cos x dx= sqrt""2 int_(0)^(pi//4) f(cos 2x) cos x dx

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