Integral calculus

posted by Febby

Please can anyone help with the following problems - thanks.

1) Integrate X^4 e^x dx

2) Integrate Cos^5(x) dx

3) Integrate Cos^n(x) dx

4) Integrate e^(ax)Sinbx dx

5) Integrate 5xCos3x dx

The standard way to solve most of these integrals is using partial integration. So, look up partial integration in your book or on the web.

Perhaps my time here is better spend by demonstrating how you can solve all these integrals using an alternative method that is often much faster.

1) Integrate x^4 e^x dx

I would do this as follows. Instead of x^4 e^x integrate e^(a x):

Integral Exp(a x) dx = Exp[a x]/a

And now we differentiate both sides four times w.r.t. a. You can interchange integration over x and differentiation w.r.t. a to obtain:

Integral x^4 Exp(a x) dx =
[24/a^5 - 24x/a^4 + 12 x^2/a^3
-4 x^3/a^2 + x^4/a] Exp(a x)

All you have to do now is put a = 1. Note that we don't bother about the integration constant here, you can add that to the result you obtain the end.

The integration constant you could have added at the beginning doesn't drop out when you differentiate w.r.t. a, because the most general integration constant is an arbitrary function of a.

2) Integrate Cos^5(x) dx

We write:

Cos^5(x) = cos(x) (1-sin^2(x))^2=

Substitute sin(x) = y, then
cos(x)dx = dy

Respond to this Question

First Name

Your Answer

Similar Questions

  1. calculus help please

    dy/dx = 2y^2 Integrating...y=2/3 y^3 + C put 1,-1 into the equation, and solve for C. Then find the y for x=2 if y= a^u see exponential functions. dy/dx=2y^2 and if y=-1 when x=1, then when …
  2. Integral calculus

    Please do help solve the followings 1) Integrate e^4 dx 2) Integrate dx/sqrt(90^2-4x^2) 3) Integrate (e^x+x)^2(e^x+1) dx 4) Integrate xe^x2 dx e^4 is a constant. 3) let u= e^x + x du= (e^x + 1)dx 4) let u= x du=dx v= e^x dv= e^x dx
  3. calculus

    1) Integrate Cos^n(x) dx 2) Integrate e^(ax)Sinbx dx 3) Integrate (5xCos3x) dx I Will be happy to critique your thinking on these. 1) Derive a recursive relation. 2) Simplest by replacing sin(bx) by Exp[i b x] and taking imaginary …
  4. integrate

    integrate du/3u and integrate x/x^2 dx it might be simple...i just need a head start...tnx
  5. PLZ integrate

    integrate cos^3x dx tnk U
  6. Calculus

    How do you integrate [(x^2)(cos(2(x^3)))]?
  7. Calculus

    6.] Replace the integral in exercise 5 (int. (1/ 1 – t) dt a = 0, b = 1/2with ?
  8. calculus 2

    Justify, with a written explanation or a mathematical reasoning and with a sketch of at least two different cases, the following properties of integrals: a) If f(x) is less than or equal to g(x) for a<=x<=b then integrate from …
  9. Calc 1

    integrate from 0 to pi/4 (sec^2x)/((1+7tanx)^2)^1/3 integrate form pi^2/36 to pi^2/4 (cos(x^1/2))/(xsin(x^1/2))^1/2 integrate from 0 to pi/3 (tanx)/(2secx)^1/2
  10. Mathematics-Integration

    Question: Prove that [integrate {x*sin2x*sin[π/2*cos x]} dx] /(2x-π) } from (0-π) = [ integrate {sin x*cos x*sin[π/2*cos x} dx ] from (0-π). My thoughts on the question: We know that integrate f(x) dx from (0-a) = integrate f(a-x) …

More Similar Questions