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Evaluate the following : int(0)^(pi//...

Evaluate the following :
`int_(0)^(pi//4)sin 3x sin 2x dx`

Text Solution

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`int_(0)^(pi//4)sin3x sin 2x dx=(1)/(2)int_(0)^(pi//4)2 sin 3x sin 2x dx`
`=(1)/(2)int_(0)^(pi//4)[cos(3x-2x)-cos(3x+2x)]dx`
`=(1)/(2)[int_(0)^(pi//4)cos x dx-int_(0)^(pi//4)cos 5xdx]`
`=(1)/(2)[sinx]_(0)^(pi//4)-(1)/(2)[(sin5x)/(5)]_(0)^(pi//4)`
`=(1)/(2)(sin.(pi)/(4)-sin0)-(1)/(10)(sin.(5pi)/(4)-sin0)`
`=(1)/(2)((1)/(sqrt2)-0)-(1)/(10)(-(1)/(sqrt2)-0)`
`" ..."[because sin.(5pi)/(4)=sin(x+(pi)/(4))=-sin.(pi)/(4)=-(1)/(sqrt2)]`
`=(1)/(2sqrt2)+(1)/(10sqrt2)=(6)/(10sqrt2)=(3)/(5sqrt2).`
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