Antiderivatives of Logarithmic and Radical Functions: Can They Be Solved?

[funcalys]

Homework Statement

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

Homework Equations

X.

The Attempt at a Solution

Wolfram Alpha seem to give no answer.

 

[tiny-tim]

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

 

[funcalys]

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 ??

 

[superg33k]

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

 

[Dickfore]

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 ??

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.

 

[tiny-tim]

hi superg33k!

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

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 ​

 

[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.

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?

 

[funcalys]

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