Evaluate The Definite Integral 0 To Pi/4 Sinx+Cosx/9+16Sin2X

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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Property 1: Integration is independent of the change of variable. `int_a ^b f(x) dx = int_a ^b f(t) dt`

Property 2: If the limits of a definite integral are interchanged then its value changes. `int_a ^b f(x) dx = – int_b ^a f(x) dx`

Property 4: If `f(x)` is a continuous function on `` then `int_a ^b f(x) dx = int_a ^b f(a+b-x) dx`

Property 5: If `f(x)` is a continuous function defined on `<0,a>` then `int_0 ^a f(x) dx = int_0^a f(a-x) dx`

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