A Limit Function Made Continuous

Difficulty: Hard · Topics: Calculus, Integration, Limits

Uzluksiz qilingan limit funksiya

The function

f(x)=begincases lim n→∞frac2x^n+1+3x^22x^n+2x+1 & (x≠ 1)\\[8pt] k/3 & (x=1)endcases

is continuous on the whole set of real numbers.

Find the value of the definite integral ∫ 0^kf(x) dx. (Here k is a constant.)