isi-entrance 2007 Q3

isi-entrance · India · solved Indefinite & Definite Integrals Piecewise/Periodic Function Integration
Show that $\int_1^n [u]([u]+1)f(u)\,du = 2\sum_{i=1}^{[n]} i \int_i^{i+1} f(u)\,du$ (or an equivalent integral identity involving the floor function).
$$\int_1^n [u]([u]+1)f(u)\,du = \int_1^2 1\cdot 2\, f(u)\,du + \int_2^3 2\cdot 3\, f(u)\,du + \cdots + \int_{[n]}^n [n]([n]+1)\,f(u)\,du$$ $$= 2\left\{1\int_1^2 f(u)\,du + 3\int_2^3 f(u)\,du + \cdots + \frac{[n]([n]+1)}{2}\int_{[n]}^n f(u)\,du\right\}$$ 
Show that $\int_1^n [u]([u]+1)f(u)\,du = 2\sum_{i=1}^{[n]} i \int_i^{i+1} f(u)\,du$ (or an equivalent integral identity involving the floor function).
Paper Questions
Q1 Q3 Q5 Q7 Q8 Q10