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Evaluate int(0)^(pi//4)(sinx+cosx)/(9+16...

Evaluate `int_(0)^(pi//4)(sinx+cosx)/(9+16sin 2x)dx`.

Text Solution

Verified by Experts

The correct Answer is:
`[(1)/(20)(log3)]`
Let `I= int_(0)^(pi//4) ((sin x + cos x))/(9+ 16 sin 2 x)dx`
` I= int _(0)^(pi//4)(sin x + cos x)/(25- 16 (sin x - cos x)^(2))dx`
Put `4 ( sin x - cos x ) = t rArr 4 (cos x + sin x ) dx = dt`
`:. I= (1)/(4) int_(-4)^(0)(dt)/(25- t ^(2))= (1)/(4)* (1)/((5))log [ |(5+t)/(5-t)|]_(-4)^(0)`
`I= (1)/(40) [ log |(5+0)/(5-0)| - log |(5-4)/(5+4)|]`
`=(1)/(40) ( log 1 - " log " (1)/(9))=(1)/(40) log 9 = (1)/(20) (log 3)`
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