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Evaluate: int0^(pi/4)(sinx+cosx)/(9+16si...

Evaluate: `int_0^(pi/4)(sinx+cosx)/(9+16sin2x)dx`

Text Solution

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`l=int_(0)^((pi)/(4))(sinx+cosx)/(9+16[1-(sinx-cosx)^(2)])dx`
`=int_(0)^((pi)/(4))(sinx+cosx)/(25-16(sinx-cosx)^(2))dx`
Let `4(sinx-cosx)=t`, so that `4(cosx+sinx)dx=dt`. Also limits `x=0` and `(pi)/(4)` are changed to `t=-4` and `0` respectively
`:.l=(1)/(4)int_(-4)^(0)(dt)/(25-t^(2))`
`=(1)/(4)int_(-4)^(0)(dt)/(5^(2)-t^(2))`
`=(1)/(4)xx(1)/(2xx5)*[ln.(5+t)/(5-t)]_(-4)^(0)`
`=(1)/(40)[ln1-ln.(1)/(9)]`
`=(1)/(40)ln9=(2ln3)/(40)=(ln3)/(20)`
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