- #1
squenshl
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Homework Statement
How do I solve ##\frac{20\ln{(t)}}{t}##?
Homework Equations
The Attempt at a Solution
Is it easier to calculate this without integrating by parts?
I'm not sure where to start.
Solving by integration by parts is easy - very few lines. I can't think of an easier way.squenshl said:Is it easier to calculate this without integrating by parts?
fresh_42 said:You have ##f\cdot f'## which is half of ##(f^2)'##. Done.
##f^2## is the solution after integration, but I said ##f \cdot f' = \frac{1}{2} \cdot (f^2)'## for the integrand. Of course this all is a bit sloppy: no integration boundaries or the constant, no mentioning of ##\ln |x|## in the OP and no ##dt 's##.Tallus Bryne said:I like the simplicity, but shouldn't it be just half of ##(f^2)## ?
edited: forgot the "half of"
Agreed. Thanks for clearing that up.fresh_42 said:##f^2## is the solution after integration, but I said ##f \cdot f' = \frac{1}{2} \cdot (f^2)'## for the integrand. Of course this all is a bit sloppy: no integration boundaries or the constant, no mentioning of ##\ln |x|## in the OP and no ##dt 's##.
Thanks everyone. The solution is ##10\ln{(t)}^2##.Math_QED said:Substitute ##u = \ln t, du = \frac {1}{t}dt##
You might like to remove an ambiguity.squenshl said:Thanks everyone. The solution is ##10\ln{(t)}^2##.
Integration is a mathematical concept that involves finding the area under a curve. It is the inverse operation of differentiation, and it is used to solve a variety of problems in physics, engineering, economics, and other fields.
Solving tricky integrals requires a combination of knowledge of integration techniques and problem-solving skills. It is important to understand the properties of integrals, such as linearity and the fundamental theorem of calculus, and to be familiar with common integration techniques, such as substitution and integration by parts.
This specific integral is used to calculate the total cost of production in certain economic models, as well as to calculate the natural logarithm and its derivatives in mathematical equations.
One tip for solving this integral is to use the substitution method, where you replace the variable t with a new variable u. You can also try to simplify the integrand by factoring out common terms or using algebraic manipulation. Additionally, using integration by parts may be helpful in solving this integral.
While some calculators have integral functions, it is important to understand the concepts and techniques behind integration in order to solve tricky integrals effectively. It is recommended to use a calculator as a tool to check your work, but not rely on it entirely.