Integration from first principles

[bigevil]

Please help me check my solution. Is there a simpler way to do this?

Homework Statement

Integrate x^3 cos x from first principles.

Homework Equations

Taylor series of cos x
Sum of powers from 1 to n

The Attempt at a Solution

I am dividing the area under the curve from a to b into n strips and then summing up the areas (where the area is x.f(x)). Then express this in terms of n and let n tend to infinity.

Let Sn = (Sigma) f(x') (delta)x
where
f(x) = x^3 cos x
f(x) = x^3 - x^5/2! + x^7/4!

x' = j . (delta)x

Insert values from 1 to n, and then group
...
...

Sn = [(delta)x]^4 . (1^3 + 2^3... + n^3)
+ [(delta)x]^6 . (1/2!) . (1^5 + 2^5 ... + n^5)
+ [(delta)x]^8 . (1/4!) . (1^7 + 2^7 ... + n^7)

Delta x = b / n, where b is the upper limit for integration

which yields 1/4 b^4 + 1/12 b^6...

 

[mace2]

I'm not really sure what "first principles" means, but shouldn't Delta x = (b-a)/n instead of just b/n?

Also you could do it with integration by parts, but I don't know if they want that or not.