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Question
Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`
Solution
Let I = `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx` ...(1)
We use the property, `int_a^b f(x)*dx = int_a^bf(a + b - x)*dx`.
Here `a = pi/(5), b = (3pi)/(10)`.
Hence changing x by `pi/(5) + (3pi)/(10) - x`, we get,
I = `int_(pi/5)^((3pi)/(10)) (sin(pi/5 + (3pi)/(10) - x))/(sin(pi/5 + (pi)/(10) - x) + cos(pi/5 + (3pi)/(10) - x))*dx`
= `int_(pi/5)^((3pi)/(10)) (sin(pi/2 - x))/(sin(pi/2 - x) + cos(pi/2 - x))*dx`
= `int_(pi/5)^((3pi)/(10)) cosx/(cosx + sinx)*dx` ...(2)
Adding (1) and (2), we get,
2I = `int_(pi/5)^((3pi)/(10)) sinx/(sinx + cosx)*dx + int_(pi/5)^((3pi)/(10)) cosx/(cosx + sinx)*dx`
= `int_(pi/5)^((3pi)/(10)) (sinx + cosx)/(sinx + cosx)*dx`
= `int_(pi/5)^((3pi)/(10))1*dx = [x]_(pi/5)^((3pi)/10)`
= `(3pi)/(10) - pi/(5)`
= `pi/(10)`
∴ I = `pi/(20)`.
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