Evaluate `int_(pi//5)^(3pi//10)(sinx)/((sinx+cosx))dx`.

Explanation

Let `I=int_(pi//5)^(3pi//10)(sinx)/((sinx+cosx))dx` ……….`(i)`
Here, `a=(pi)/(5)` and `b=(3pi)/(10)implies(a+b)=((pi)/(5)+(3pi)/(10))=(pi)/(2)`.
Using `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx`, we have
`I=int_(pi//5)^(3pi//10)(sin((pi)/(2)-x))/(sin((pi)/(2)-x)+cos((pi)/(2)-x))dx`
`impliesI=int_(pi//5)^(3pi//10)(cosx)/((sinx+cosx))dx`.......`(ii)`
Adding `(i)` and `(ii)` , we get
`2I=int_(pi//5)^(3pi//10)dx=[x]_(pi//5)^(3pi//10)=((3pi)/(10)-(pi)/(5))=(pi)/(10)`
`impliesI=(pi)/(20)`.