Deriving the Integral via Leibniz Rule

In summary, the conversation discusses the solution to the integral \int_0^1\frac{x-1}{\ln{x}} dx and its relationship to \Phi(\alpha)=\int_0^1\frac{x^{\alpha}-1}{\ln{x}} dx. The answer provided a derivative of \Phi(\alpha), but there was a mistake in the calculation. The correct derivative is \Phi'(\alpha)=\frac{1}{\alpha+1}.
  • #1
dirk_mec1
761
13

Homework Statement


[tex]
\int_0^1\frac{x-1}{\ln{x}} dx
[/tex]

Homework Equations


[tex]
\Phi(\alpha)=\int_0^1\frac{x^{\alpha}-1}{\ln{x}} dx
[/tex]

The Attempt at a Solution


In the answers they say:[tex]
\Phi '(\alpha)=\int_0^1\frac{x^{\alpha}\ln{x}}{\ln{x}} dx=\frac{1}{\alpha+1}
[/tex]but the derative is wrong, right? I don't understand how they calculated the derative...
 
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  • #2
The derivative is d/d(alpha). x^alpha=e^(log(x)*alpha). Actually, it is right.
 
  • #3
Dick said:
The derivative is d/d(alpha). x^alpha=e^(log(x)*alpha). Actually, it is right.

You're right I accendentally differentiated w.r.t x, thanks Dick.
 

Related to Deriving the Integral via Leibniz Rule

1. What is Integration via Leibniz?

Integration via Leibniz, also known as the Leibniz integral rule, is a method for finding the derivative of a definite integral with a variable upper limit. It was developed by the mathematician Gottfried Wilhelm Leibniz in the 17th century.

2. How does Integration via Leibniz work?

The Leibniz integral rule states that for a definite integral with a variable upper limit, the derivative of the integral is equal to the integrand evaluated at the upper limit multiplied by the derivative of the upper limit. This can be written as d/dx ∫f(x,t)dt = f(x,b) * d/dx(b).

3. What are some practical applications of Integration via Leibniz?

Integration via Leibniz is commonly used in physics and engineering to solve problems involving rates of change and motion. It is also used in economics and finance to calculate marginal changes and optimization problems.

4. Are there any limitations to Integration via Leibniz?

Integration via Leibniz can only be applied to definite integrals with a variable upper limit. It cannot be used for indefinite integrals or integrals with a fixed upper limit. Additionally, it may not always be the most efficient method for finding derivatives of integrals.

5. Can Integration via Leibniz be generalized to higher dimensions?

Yes, the Leibniz integral rule can be extended to higher dimensions, known as the multivariate Leibniz rule. It is used to find partial derivatives of integrals with multiple variables and is a fundamental tool in multivariable calculus.

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