How calculate limit of string if i don't know calculate integral? string u=n^4*integral x/1+x^5 integral is definite in [n,n+1]class tutor said that i need not to know integrals

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The Mean Value Theorem helps you to solve the problem.

The function under the definite integral is a continuous function, therefore the Mean Value Theorem is applicable.

`int_a^bf(x)dx=(b-a)*f(c)` , c`in` [a,b]

In this case, the string=`n^4` *`int_n^(n+1)x/(1+x^5)dx` `=n^4*(n+1-n)*f(c_n)=n^4*f(c_n)`

Use the notation `u_n`  for the string.

`u_n=n^4*c_n/(1+(c_n)^5)`

`u_n=(n/c_n)^4*(c_n)^5/(1+(c_n)^5)`

Use`nltc_nltn+1`  => `lim_(n->oo)``c_n=oo`

`1/ngt1/c_ngt1/(n+1)`

`` Multiply by n>0 =>

`n/ngtn/c_ngtn/(n+1)=gt1gtn/cgtn/(n+1)` =>`lim_(n->oo)` `(n/c_n)=1`

ANSWER: The limit of string is 1.

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