# Calculate the antiderivatives

1. Nov 3, 2012

### funcalys

1. The problem statement, all variables and given/known data
$\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$

2. Relevant equations
X.

3. The attempt at a solution
Wolfram Alpha seem to give no answer.

Last edited by a moderator: Jun 19, 2014
2. Nov 3, 2012

### tiny-tim

the x2 + 4x makes that impossible to do by analytic means

3. Nov 3, 2012

### funcalys

Re: $\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$

Actually my original problem was determining the convergence or divergence of the following improper integral:
$\int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$
I split the integral into
$\int^{+∞}_{1} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$
and $\int^{1}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$
, calculate the antiderivatives, then evaluate the limit of them.
So if I can't calculate the antiderivative, is there any alternative way to see if this integral convergent or not ??

4. Nov 3, 2012

### superg33k

Re: $\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$

Tiny-Tim, how did you know thats impossible to do by analytical means? What should I google to learn more?

5. Nov 3, 2012

### Dickfore

Do you know how to find the asymptotic behavior of your integrand for $x \rightarrow \infty$ and $x \rightarrow 0$? If yes, then you may use the comparison test.

6. Nov 4, 2012

### tiny-tim

hi superg33k!
it's obvious just from looking at it … that bracket is simply too complicated for any of the known techniques to work!

once you've had lots of practice at differentiating and integrating, you'll see why

7. Nov 4, 2012

### Dickfore

I'll start off. Let us consider the upper bound first. For $x \rightarrow \infty$. $x^2 = o(4^x)$, so the logarithm in the numerator behaves as $\sim x \, \ln(4)$. Similarly, $7 x^3 = o(3 x^7)$, so the expression under the square root in the denominator behaves as $\sim 3 x^7$. Therefore, the integrand behaves as:
$$\sim \frac{x \, \ln(4)}{\sqrt{3 x^7}} = \frac{\ln(4)}{\sqrt{3}} \, x^{-5/2}$$
Do you know whether the integral:
$$\int_{1}^{\infty}{x^{-5/2} \, dx}$$
is convergent or divergent?

A similar analysis can be done on the lower bound of the integral. However, what are the dominant terms in this limit?

8. Nov 4, 2012

### funcalys

Re: $\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx$

Thank you very much, I can take it from here :D.