- #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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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 ??
superg33k said:Tiny-Tim, how did you know that's impossible to do by analytical means? What should I google to learn more?
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.
An antiderivative is the inverse operation of taking a derivative. It is a function that, when differentiated, gives back the original function.
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.
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.
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.
Some common antiderivatives include power functions, trigonometric functions, logarithmic functions, and exponential functions.