I have a question regarding integrals, which I have not been able to solve

Question

The function to be integrated is the following:

f(x)=(√x)/(x² + ln(1+x))

 

In addition to this we were asked to confirm if the function converges or not. When plotted it is observed to tend towards 0 as x approaches infinity, and a calculation of limits will confirm this. But I can't solve the integration. 

 

Answer 1

 

Let  u=(x)^½ and v=x²+ln(1+x)   then use the quotient rule:

(many steps later):

 

f'(x) = [(x+1)ln(x+1) - x(3x²+3x+2) ] / (2x^½*[(x+1)(x²+ln(x+1)]²)

 

On convergence as x—>∞:

  The numerator —>∞

  The denominator —>∞

So you can apply L'Hospital's (take derivative of numerator and denominator separately):

  Lim(x—>∞) { ½x^-(½) / (2x + 1/(1+x)) } = 0