- calculus - Integration by Parts Question: Integrate $x^3e^x . . .
A nice and quick way to visualize integration by parts (it could be a time-saver!): $$\matrix { \text {differentiate} \text {integrate} \\ x^3 e^x
- Find the general solution to $xy = 2y + x^3e^x$
Find the general solution to $xy' = 2y + x^3e^x$ Ask Question Asked 4 years ago Modified 4 years ago
- How to arrange $e^3,3^e,e^ {\pi},\pi^e,3^ {\pi},\pi^3$ in the . . .
I don't understand what you are asking What do you mean by size of the order? Do you mean you want to order the numbers by size?
- Use Laplace Transformations to solve $y+2y+5y=3e^{-x}sin(x)$, with . . .
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges,
- calculus - (Laplace Method) $y - 4y = 6e^ {3t} - 3e^ {-t . . .
Taking the Laplace Transform to both sides of the ODE we get \begin {align*} s^2Y (s)-s (1)- (-1)-4 [sY (s)-1] =\frac 6 {s-3}-\frac 3 {s+1}\\ [4pt] (s^2-4s)Y (s
- Integral of $\int z^3 e^z$ - Mathematics Stack Exchange
However, what you’ve written is incorrect, starting with the missing $dz$ in the integral: $\int z^3e^z\,\underline {dz}$ For your integration by parts $u=z^3$, not $z^3\,dz$, and $dv=e^zdz$, not $e^z$: differentials can only be equal to other differentials
- Solving $\int_0^ {\infty}x^3e^ {-x^2}dx$ [duplicate]
You used a substitution, and then integration by parts once But what if you start out with integration by parts? $$ \begin {align} \int x^3e^ {-x^2}\,dx =\int -\frac12x^2\left (-2xe^ {-x^2}\right)\,dx\\ =-\frac12x^2e^ {-x^2}+\int xe^ {-x^2}\,dx\\ =-\frac12x^2e^ {-x^2}-\frac12e^ {-x^2}+C \end {align} $$ And "from $0$ to $\infty$ " gives
- How many ways to evaluate $1 + e^ {-x} + e^ {-2x} + e^ {-3x} + \ldots . . .
Geometric series: $$1+e^ {-x}+e^ {-2x}+\ldots=\sum_ {n=0}^\infty\left (e^ {-x}\right)^n=\frac1 {1-e^ {-x}}=\frac {e^x} {e^x-1}$$ as long as $\;e^ {-x}<1\iff e^x>1\iff
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