Integral involving the derivative

In summary, the conversation was about calculating the integral \int \frac{f'(x)}{f^2(x)}dx and the expert suggests rewriting it as \int f^{-2}\,df and using the power rule to solve it. They also provide an alternative solution using the substitution rule by letting u=f(x) and using u^{-2}\,du.
  • #1
Yankel
395
0
Hello,

I am trying to calculate the following integral:

\[\int \frac{f'(x)}{f^{2}(x)}dx\]

I suspect is has something to do with the rule of f'(x)/f(x), with the ln, but there must be more to it than that.

can you assist please ? Thank you !
 
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  • #2
I would rewrite the given integral as:

\(\displaystyle \int f^{-2}\,df\)

Now just use the power rule. :)
 
  • #3
Can you explain your solution please ? How did you go from the derivative and dx to f only with df ?
 
  • #4
\(\displaystyle \int \frac{f'(x)}{f^2(x)}\,dx=\int f^{-2}\d{f}{x}\,dx=\int f^{-2}\,df\)
 
  • #5
Right. Simple...

Could you get the same result from the substitution rule ?
 
  • #6
Yankel said:
Right. Simple...

Could you get the same result from the substitution rule ?

Yes, I suppose we could write:

\(\displaystyle u=f(x)\implies du=f'(x)\,dx\)

And now we have:

\(\displaystyle \int u^{-2}\,du\)

That seems a little roundabout to me, but perfectly legitimate. :D
 

FAQ: Integral involving the derivative

What is an integral involving the derivative?

An integral involving the derivative is a mathematical expression that involves both the integral and derivative operations. It is commonly used to solve problems where the rate of change of a quantity needs to be determined over a certain interval.

What is the purpose of using an integral involving the derivative?

The purpose of using an integral involving the derivative is to find the original function when only its derivative is known, or to determine the change in a quantity over a given interval. This is useful in many real-world applications, such as calculating the velocity of an object from its acceleration.

How do you solve an integral involving the derivative?

To solve an integral involving the derivative, you can use the fundamental theorem of calculus, which states that the integral of a function's derivative is equal to the original function. This allows you to use integration techniques to solve for the original function.

Are there any specific rules or formulas for solving integrals involving derivatives?

Yes, there are specific rules and formulas for solving integrals involving derivatives. Some common rules include the power rule, product rule, and chain rule. Additionally, the use of substitution and integration by parts may also be necessary in more complex cases.

Can an integral involving the derivative be evaluated using numerical methods?

Yes, an integral involving the derivative can be evaluated using numerical methods, such as the trapezoidal rule or Simpson's rule. These methods involve approximating the integral using smaller intervals and calculating the sum of these smaller areas to get an estimate of the integral's value.

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