The limits of integration are: 1 to 2
The function to integrate is: (2x^3 + 1)/(x^4 + 2x)^3
You should note that (2x^3 + 1) is a differential coefficient of (x^4 + 2x)
i.e when you differentiate (x^4 + 2x) you get 4x^3 + 2 = 2(2x^3 + 1)
So ignore the differential coefft bit and just look at 1/(x^4 + 2x)^3 = (x^4 + 2x)^(-3)
When we "integrate" (x^4 + 2x)^(-3), we get (x^4 + 2x)^(-2)
Now differentiate (x^4 + 2x)^(-2) to get (-2)(4x^3 + 2)(x^4 + 2x)^(-3) = (-4)(2x^3 + 1)(x^4 + 2x)^(-3)
So now we simply divide the (x^4 + 2x)^(-2) by (-4) to get our integral, which is (-1/4)(x^4 + 2x)^(-2).
Now take the limits,
[(-1/4)(x^4 + 2x)^(-2)][1 .. 2] = {(-1/4)(16 + 4)^(-2) - (-1/4)(1+2)^(-2)} = (-1/4){1/20^2 - 1/3^2)
= (1/4){1/9 - 1/400} = (1/4){(400-9)/3600} = 391/14400
Answer: 391/14400