- #1

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## Homework Statement

[itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

## Homework Equations

X.

## The Attempt at a Solution

Wolfram Alpha seem to give no answer.

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- Thread starter funcalys
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- #1

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[itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

X.

Wolfram Alpha seem to give no answer.

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- #2

tiny-tim

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the x^{2} + 4^{x} makes that impossible to do by analytic means

- #3

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Actually my original problem was determining the convergence or divergence of the following improper integral:

[itex]\int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

I split the integral into

[itex]\int^{+∞}_{1} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

and [itex]\int^{1}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

, 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

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

- #5

- 2,967

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[itex]\int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

I split the integral into

[itex]\int^{+∞}_{1} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

and [itex]\int^{1}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

, 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 [itex]x \rightarrow \infty[/itex] and [itex]x \rightarrow 0[/itex]? If yes, then you may use the comparison test.

- #6

tiny-tim

Science Advisor

Homework Helper

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it's obvious just from looking at it … that bracket is simply

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

- #7

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

I'll start off. Let us consider the upper bound first. For [itex]x \rightarrow \infty[/itex]. [itex]x^2 = o(4^x)[/itex], so the logarithm in the numerator behaves as [itex]\sim x \, \ln(4)[/itex]. Similarly, [itex]7 x^3 = o(3 x^7)[/itex], so the expression under the square root in the denominator behaves as [itex]\sim 3 x^7[/itex]. Therefore, the integrand behaves as:

[tex]

\sim \frac{x \, \ln(4)}{\sqrt{3 x^7}} = \frac{\ln(4)}{\sqrt{3}} \, x^{-5/2}

[/tex]

Do you know whether the integral:

[tex]

\int_{1}^{\infty}{x^{-5/2} \, dx}

[/tex]

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

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

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