Evaluate Integrals??? How do you do them???

Question

Forinstance, integral

x dx
------------
(7x^2+3)^5 ?????

Answer 1

I = ∫ x dx / (7x ² + 3)^5
Let u = 7x ² + 3
du / dx = 14x
du = 14x dx
du /14 = x dx
I = (1/14) ∫ u^(- 5) du
I = (1/14) u^(-4) / (- 4) + C
I = - 1 / [ 56 (7x ² + 3)^4 ] + C

Answer 2

Integral set-up is sort of like a bag of tricks.
One que is that you have a udu type setup.
Suppose we set u= 7x^2+3, then du = 14x
While we don't have 14 x, we have x. So we can TRANSFORM this to
xdx = du/14
denominator = u^5
Then the integration is (1/14) u**-5 du , then back transform to the result in term of x.