Integrating my head P1

Let $f : \mathbb N \to \mathbb N$ such that $\displaystyle f(n) = \int_0^n \left( k^{k/2} + (k-1)^{k/3} \right) \, dk $.

Can $\displaystyle \sum_{n=i}^j f(n)$ be an integer for some $(i,j) \in \mathbb N, i < j$ ?

I will be posing sets of 6 problems, as a challenge for people to solve.

#Calculus

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

I don't understand. You're integrating with respect to what variable?

Pi Han Goh - 2 years, 1 month ago

Uh, with respect to k. The following picture was from Word, and hence the equation did not have an option to have the integration from k=0, so, it just showed 0. Sorry for any inconveniences.

Alex Fullbuster - 2 years, 1 month ago

That still does not make sense, because there's no $n$ in the right hand side of the equation $f(n) = \cdots$ .

Pi Han Goh - 2 years, 1 month ago

Yes, I noticed that, again error from word. I am reposting the photo from a earlier version of Word where there are no mistakes. Thanks for informing sir.

Alex Fullbuster - 2 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$