what is f when f'(x)=(x^2=2x)/(sqrt x^3+3x^2+1) and f(0)=1

Question

what is f when f'(x)=(x^2=2x)/(sqrt x^3+3x^2+1) and f(0)=1

Answer 1

I will assume you meant the numerator in this problem to read x^2+2x. The problem is solved via substitution.

Let u = x^3+3x^2+1, then du/dx = 3x^2 +6x which is exactly 3 times the numerator. 

So the problem becomes

1/3 (integral sign) du/sqrt(u) which has anti-derivative by power rule to be 2/3 * sqrt(u) + c.

Plug u back in and solve for c with point given. I come up with 2/3 sqrt(x^3 +3x^2+ 1) + 1/3.