Solve Difficult Integrals Step-by-Step

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SUMMARY

This discussion focuses on solving complex integrals, specifically the integral from 1 to infinity of 1 / (x ln(∛x)) dx and the integral of sqrt(1-x)/sqrt(x) dx. The first integral requires advanced techniques in improper integrals, while the second can be simplified using the substitution x = sinh²(t). The second integral's result is expressed as 1/(x ln(x^(1/3))) = 3/(x ln(x)), indicating a relationship with derivatives of logarithmic functions.

PREREQUISITES
  • Understanding of improper integrals
  • Familiarity with substitution methods in integration
  • Knowledge of logarithmic functions and their properties
  • Basic calculus concepts, including derivatives and integrals
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  • Study techniques for evaluating improper integrals
  • Learn about substitution methods in integral calculus
  • Explore the properties of logarithmic functions in calculus
  • Investigate the relationship between integrals and derivatives
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Students, mathematicians, and educators seeking to deepen their understanding of integral calculus, particularly those tackling advanced integration techniques.

poopforfood
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Please Help Me!

Hello all I need some serious help with these problems...If you could help me solve them step by step that would be fantastic

Evaluate the following integrals


Integral from 1 to Infinity of 1 / xln(thirdrootx) dx

Integral of sqrt(1-x)/sqrt(x) dx
 
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[tex]\frac{1}{x \ln(x^{\frac{1}{3}})} = \frac{3}{x \ln(x)}[/tex]

This is an exact derivative of a function. Can you work out what it is?
 
As for the second one, try letting [itex]x=\sinh^2 t[/itex] =] Its quite nice after that
 

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