Given: I=nπ+λ∫0|sinx|dx n∈N, 0≤λ<π
⇒I=λ∫0|sinx|dx+nπ+λ∫λ|sinx|dx
Using the property:
a+nT∫af(x) dx=nT∫0f(x) dx
Where T is the period of f(x)
For |sinx|, the period is π
Now,
⇒I=λ∫0|sinx|dx+nπ∫0|sinx|dx =[−cosx]λ0+nπ∫0sinx dx =−(cosλ−1)+n(1+1)∴I=2n+1−cosλ