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

In summary, the conversation is about determining the convergence or divergence of the improper integral \int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx and finding alternate methods to solve it if the antiderivative cannot be calculated. Tiny-Tim suggests using the comparison test by finding the asymptotic behavior of the integrand for x \rightarrow \infty and x \rightarrow 0. Superg33k provides a breakdown of the asymptotic behavior for x \rightarrow \infty and suggests continuing the analysis for the lower bound.
  • #1
funcalys
30
1

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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  • #2
the x2 + 4x makes that impossible to do by analytic means
 
  • #3


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


Tiny-Tim, how did you know that's impossible to do by analytical means? What should I google to learn more?
 
  • #5
funcalys said:
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 ??

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
hi superg33k! :smile:
superg33k said:
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! :redface:

once you've had lots of practice at differentiating and integrating, you'll see why :smile:
 
  • #7
Dickfore said:
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


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

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

What is an antiderivative?

An antiderivative is the inverse operation of taking a derivative. It is a function that, when differentiated, gives back the original function.

Why do we need to calculate antiderivatives?

Antiderivatives are useful in many areas of science and engineering, particularly in physics and economics, as they allow us to find the original function from its rate of change.

How do we calculate antiderivatives?

To calculate an antiderivative, we use a set of rules and formulas known as integration techniques. These techniques involve manipulating the function in order to find its antiderivative.

What is the difference between definite and indefinite antiderivatives?

A definite antiderivative has specific limits of integration, while an indefinite antiderivative does not. In other words, a definite antiderivative gives a specific numerical value, while an indefinite antiderivative gives a general formula.

What are some common antiderivatives?

Some common antiderivatives include power functions, trigonometric functions, logarithmic functions, and exponential functions.

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