Using integral estimates, for instance, you can easily see that $f (n)$ should grow asymptotically like $n^3/3$.

For these six numbers, e3, 3e, eπ, πe, 3π, π3, how to arrange them in the increasing order? This problem is taken from the today test: National Higher Education Entrance Examination.

Nov 17, 2014 · $$5+55+555+5555+\cdots+\overbrace {55\dots5}^ {n\text { fives}}$$ $$=\frac59 (9+99+999+9999+\cdots+\overbrace {99\dots9}^ {n\text { nines}})$$ $$=\frac59 (10^1-1+10^2 ...

Aug 14, 2015 · A nice and quick way to visualize integration by parts (it could be a time-saver!): $$\matrix {&\text {differentiate}&&& &\text {integrate}&\\ &x^3&&&&e^x ...

If you simplify by factoring out $e^ {2t}$ and cancelling, that would give you $$g' - 2g = 3e^ {2t},$$ instead of $g'-2g = 3e^ {4t}$, which is what you had. So it looks like you made a simplification …

I need to find the value of $\displaystyle \int _0^ {2\pi}\sin^2 \left (\frac {-\pi} {6}+3e^ {it} \right)dt$. I figured I could use contour integration and the Cauchy-Goursat theorem to do so.

Mar 12, 2020 · What is the exact value of 3.199 E +01 ? Does it mean $3.199\times10^1$?? I get confused with the notation.

$$A (2t\cdot 3e^ {3t}+t^2\cdot 9e^ {3t}+2e^ {3t}+6te^ {3t}+9 (At^2e^ {3t}) = t^2e^ {3t}$$ but how should I group the terms in the left to equate to the ones in the right?

However, what you’ve written is incorrect, starting with the missing $dz$ in the integral: $\int z^3e^z\,\underline {dz}$. For your integration by parts $u=z^3$, not $z^3\,dz$, and …

Sep 9, 2024 · Solving $\int_0^ {\infty}x^3e^ {-x^2}dx$ [duplicate] Ask Question Asked 1 year, 3 months ago Modified 1 year, 3 months ago