Question

Integration by parts.

integral of xe^(-5x) dx

Answer 1

u = x
du = dx

dv = e^(-5x)
v = (-1/5)x e^(-5x)?

Answer 2

I'm not familiar with your notation, but I think you're on the right track.

Answer 3

The standard way is use the formula for integration by parts.

integral u(x)v(x) dx =
u(x) integral v(x)dx - integral [ ( integral v(x)dx ) (du/dx) ] dx

Let u(x) = x and v(x) = e^5x.

However you can often do these quicker by guessing what part of the answer will be.

Knowing that the derivative of e^x is e^x, and knowing how to differentiate the product of two functions, you can guess the answer will probably contain the term like xe^5x.

d/dx (xe^5x) = 5xe^5x + e^5x

So integral xe^5x dx
= 1/5 integral (5xe^5x + e^5x - e^5x) dx
= 1/5 integral (5xe^5x + e^5x) dx - 1/5 integral e^5x dx
= 1/5 xe^5x - 1/25 e^5x + C

Answer 4

@Thomas9 & others, for reference, Integration by Parts:
\[\int\limits_{a}^{b}U \cdot dV = U \cdot V - \int\limits_{a}^{b} V \cdot dU\]

Answer 5

^yup

Answer 6

so getting back to my solving its u * v - int vdu

\[x * e^{-5x} - -\frac{1}{5}\int\limits e^{-5x} dx\]

Answer 7

\[x * e^{-5x} + \frac{1}{25} e^{-5x} + c\]

Answer 8

You're missing -1/5 in the first term.

Answer 9

oh woops yeah.

Answer 10

\[-\frac{1}{5}x * e^{-5x} + \frac{1}{25}e^{-5x} + c?\]

Answer 11

The second + is a minus.

Answer 12

first plus actually.

Answer 13

wasn't it - -1/5 intgral?

Answer 14

since initially it was a - the integral, and the dx = -1/5du?

Answer 15

You get another -1/5 from integrating in the last step.

Answer 16

oh yeah tha't sright....

Answer 17

forgot about that :). ty