Integral calculus

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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

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