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paulmdrdo1
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can someone give me hints how to solve this.(Coffee)
$\displaystyle \int\frac{ln^2\,3x}{x}dx$
$\displaystyle \int\frac{ln^2\,3x}{x}dx$
paulmdrdo said:can someone give me hints how to solve this.(Coffee)
$\displaystyle \int\frac{ln^2\,3x}{x}dx$
To add to Sudharaka's sage suggestion, whenever you have a ln in numerator and x in the denominator of the integrand, some kind of u = ln(x) substitution bears looking into.paulmdrdo said:can someone give me hints how to solve this.(Coffee)
$\displaystyle \int\frac{ln^2\,3x}{x}dx$
Integration is a mathematical process used to find the area under a curve by breaking it down into smaller, simpler parts. It is the inverse operation of differentiation and is used to solve a wide range of problems in fields such as physics, engineering, and economics.
Integration can lead to natural logarithms (ln) when solving certain types of integrals, specifically those involving exponential functions. By using a technique called substitution, we can manipulate the integral to resemble the derivative of ln, allowing us to solve it using the ln function.
The relationship between integration and ln is that ln is the inverse of the natural exponential function (e^x), and integration is the inverse of differentiation. This means that integrating an exponential function will result in an ln function, as ln "undoes" the effect of e^x.
Ln is commonly used in integration because it is a natural logarithm that uses the base e, which is a special number that appears in many mathematical and scientific formulas. Ln is also useful for solving exponential growth and decay problems, which are common in many fields.
To solve an integration problem leading to ln, you can use the substitution method or integration by parts. First, try to manipulate the integral to resemble the derivative of ln, then use the appropriate integration technique to solve it. It is also important to remember to use the integration constant (C) when solving indefinite integrals leading to ln.