Calculus antideritvative help?

Question

how do you find the antiderivative of the equation p(x)=300x/6x^2+5? I know you defiantly have to use U-substituion but i donno how...

Answer 1

Assuming +5 is part of the denominator

∫300x/(6x^2+5) dx

Let u = 6x^2 +5
Then du/dx = 12x

So dx = du/12x substitute u and du in
∫(300x du) /(12x u)

Note the x in 300x/12x cancels and leaving you with only u and du. This is what you want to happen with u substitution. Hopefully the simplified u and du form is something you know how to integrate.

∫25 du/u

bring the 25 out since it's constant
25 ∫ du/u

Integrate knowing that ∫du/u = ln |u| +c
25 ln | u | +c

replace u
25 ln | 6x^2 + 5 | + C

Answer 2

p(x) = 300x / (6x^2 + 5) is what I assume you mean, if so:

int{p(x)} = int{300x dx / (6x^2 + 5)}

Let u = 6x^2 + 5 ==> du = 12x dx
Therefore:

int{p(x)} = int{25 du / u}
int{p(x)} = 25 int{du / u}
int{p(x)} = 25 * ln(u) + C
int{p(x)} = 25*ln(6x^2 + 5) + C

Answer 3

25ln(6x^2) + c

b/c u=6x^2
25S(1/u)