Animated Solution for Mathematics - Definite Integration: Let $a_n={\int }_{-1}^n\left(1+\frac{x}{2}+\frac{x^2}{2}+\frac{x^3}{3}+\cdots +\frac{x^{n-1}}{n}\right)dx$ for $n\in \mathbb{N}$. Then the sum of all the elements of the set $\{n\in \mathbb{N}:a_n\in (2,30)\}$ is

Imagine you are standing before a complex, sprawling polynomial. It looks intimidating, but as an elite JEE aspirant, you know that complexity is often just a mask for underlying elegance.

Our goal is to find the natural numbers $n$ such that the integral falls strictly within the interval $(2,30)$.

The first step in any great mathematical journey is simplification. We condense the polynomial into compact sigma notation: ${\sum }_{k=1}^n\frac{x^{k-1}}{k}$.

This is not just a shorthand; it is a way of seeing the structure. The $k$-th term is simply $x^{k-1}$ divided by $k$.

Because integration is a linear operation, we can swap the integral and the summation. This is where the magic happens.

We are now looking at the integral of a single term: $\int \frac{x^{k-1}}{k}dx$.

Applying the fundamental power rule of calculus, we increase the exponent of $x$ by one, turning $x^{k-1}$ into $x^k$, and divide by the new exponent $k$. The $k$ already in the denominator multiplies with this new $k$, giving us $k^2$.

So, the integral of the $k$-th term is simply $\frac{x^k}{k^2}$. Our expression for $a_n$ now looks like this:

With the integration complete, we apply the limits of integration, $n$ and $-1$. Substituting the upper limit $n$ and subtracting the result of the lower limit $-1$, we arrive at the general formula:

This formula is our compass. It tells us exactly how $a_n$ behaves for any natural number $n$.

Now, we test our values:

For $n=1$: Since our interval is $(2,30)$, $2$ is not included. We reject $n=1$.

For $n=2$: This is clearly in our interval. We accept $n=2$.

For $n=3$: This is also in our interval. We accept $n=3$.

For $n=4$, the terms grow rapidly, and $a_4\approx 31.9$, which exceeds 30. We reject $n=4$. Since the sequence is strictly increasing, any $n>4$ will also be rejected.

The valid values are $n=2$ and $n=3$. The sum of these elements is $5$.