calc

posted by .

how do you start this problem:

integral of xe^(-2x)

There are two ways:

1) Integration by parts.

2) Differentiation w.r.t. a suitably chosen parameter.

Lets do 1) first. This is the "standard method", but it is often more tedious than 2)

You first write the integral as:

Inegral xe^(-2x) dx =

Integral -1/2 x d(e^(-2x))

Here we have used that:

d(e^(-2x)) = -2 e^(-2x)

The next is is to make use of the fact that:

d(f g) = f dg + g df --->

f dg = d(fg) - g df

This yields:

Integral -1/2 x d(e^(-2x)) =

Integral d[-1/2 x e^(-2x)] -

Integral -1/2 e^(-2x) dx =

-1/2 x e^(-2x) - 1/4 e^(-2x) + C


Method 2) is much simpler. Consider the function:

e^(ax)

It's integral is:

Integral e^(ax)dx = 1/a e^(ax)

Le's differentiate both sides w.r.t. a:

Integral x e^(ax)dx =

[ -1/a^2 + x/a] e^(ax)

And insert a = -2 to obtain the answer.

Personal Finance

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Stats: Joint Density Function

    Let the joint density function of X and Ybe given by: f(x)={kxy^2 for 0<x<y<1 (0 otherwise what is the value of k?
  2. Calculus

    Hello, I have some calculus homework that I can't seem to get started..at least not on the right track?
  3. integration by parts

    s- integral s ln (2x+1)dx ? = ln(2x+1)x - s x d( ln (2x+1)) = ln(2x+1)x- s x [(2x+1)'/ (2x+1)] dx = ln(2x+1)x- s x [(2)/ (2x+1)] ?
  4. calc

    evaluate the integral: y lny dy i know it's integration by parts but i get confused once you have to do it the second time Leibnitz rule (a.k.a. product rule): d(fg) = f dg + g df y lny dy = d[y^2/2 ln(y)] - y/2 dy ----> Integral …
  5. Math/Calculus

    How would I integrate the following by parts: Integral of: (x^2)(sin (ax))dx, where a is any constant. Just like you did x^2 exp(x) below. Also partial integration is not the easiest way to do this integral. You can also use this method. …
  6. calc asap!

    can you help me get started on this integral by parts?
  7. Calculus II

    Evaluate the integral using method of integration by parts: (integral sign)(e^(2x))sin(5x)dx
  8. Calculus - integration

    Integral of (2x+5sin(x))^2 dx i didn't know how to start this question so i expand the squre first and then did the rest but its wrong and i don't think my method is right. thank you.
  9. Calculus

    Use integration by parts to find the integral. Round the answer to two decimal places if necessary. (x+4) ln x dx between 3 and 0. I tried this problem two different ways and got two different answers 1.63 and 3.89. which one is correct?
  10. Calc 2

    a. Integral (x^2)/(sqrt(1+(x^2))) Would I separate these two into 2 separate integrals?

More Similar Questions