limits of sequence

its very easy
i can explain it to you via an illustration

EXAMPLE: ∫1+x dx
remember limit of a sum is just a way of approximation

Always remember the formula
∫_a^b f(x) dx= (b-a) lim_{n→∞} 1/n [f(a) + f(a+h) +f(a+2h) +...f(a+(n-1)h)
where h=b-a/n

∫₁²1+x dx =lim n→∞ 1/n [1+(1+1) + (1+2)+(1+3)+...(1+(n-1))
RHS=1+1+1...n times +(1+2+3+4+5+....(n-1 terms))
RHS =n+[(n-1)(n)/2]
The above was calculated using the sum of n natural numbers formula

RHS=take common n and then apply limit to get the answer

NOTE#In jee, application of reverse process is more important ie, converting summation limits into integrations...

This is done by replacing (summation r=1 to n) by ∫₁ⁿ
1/n by dx and r/n by dx
memorize the above as it might be helpful for u