Dear Samsul Islam, if a real function f on a closed and bounded interval [a,b] is Riemann integrable on [a,b], then f is bounded on [a,b]. Taking the contrapositive of the previous conditional, one derives that if f is unbounded on [a,b], then the Riemann integral of f on [a,b] does not exist. In other words, boundedness of f is a necessary – although not sufficient – condition for the integral of f to exist on [a,b]. Your function is unbounded on [0,2], since it diverges to positive infinity as x tends to 0+. Hence, the Riemann integral of your function on [0,2] does not exist.

Besides, the partial fraction decomposition (3x^2 + 2x + 1) / (x^3 + x^2)

= -x/(x^3) + 5/(x+1) you write is not correct; the left side of the equation is not equal to the right side of the equation.

Given any 0