# Calculate the antiderivatives

funcalys

## Homework Statement

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

X.

## The Attempt at a Solution

Wolfram Alpha seem to give no answer.

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Homework Helper
the x2 + 4x 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.

Homework Helper
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.