What is the integration of (ln(x))^n?

In summary, the integration of (ln(x))^n can be done using the integration by parts method. The formula for integrating ln x is x(ln x) - x + C, and for (ln x)^2 it is x(ln x)^2 - x ln x - (x ln x - x - x). The process can be repeated to integrate higher powers of ln x.
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what is the integration of (ln(x))^n?
 
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To integrate ln x by parts, let u= ln x, dv= dx. Then du= (1/x)dx and v= x so
[tex]\int ln x= x(ln x)- \int dx= xln x- x+ C[/tex]

To integrate [itex](ln x)^2[/itex] by parts, let u= ln x, dv= ln x dx. Then du= (1/x)dx and v= xln x- x so
[tex]x(ln x)^2- x ln x- \int ln x- 1 dx= x(ln x)^2- x ln x- (x ln x - x- x)= x(ln x)^2- 2x+ C[/tex]

Keep integrating by parts until you see a pattern.
 

Related to What is the integration of (ln(x))^n?

Question 1: What is the definition of integration?

The integration of a function is the process of finding the area under the curve of that function. It is the inverse operation of differentiation, and is used to solve problems involving rates of change and accumulation.

Question 2: What is the general formula for integrating (ln(x))^n?

The general formula for integrating (ln(x))^n is ∫(ln(x))^n dx = x(ln(x))^n - n∫(ln(x))^(n-1) dx + C, where C is the constant of integration.

Question 3: How do you solve the integration of (ln(x))^n?

To solve the integration of (ln(x))^n, you can use the general formula and apply the power rule for integration. This involves rewriting the function as (ln(x))^n = (ln(x))^n-1 * (ln(x)), and then using integration by parts to solve for the integral.

Question 4: Are there any special cases for integrating (ln(x))^n?

Yes, there are special cases for integrating (ln(x))^n. When n = 1, the integral becomes ∫ln(x) dx = xln(x) - x + C. When n = 0, the integral becomes ∫dx = x + C. And when n = -1, the integral becomes ∫(ln(x))^(-1) dx = ln(ln(x)) + C.

Question 5: Can the integration of (ln(x))^n be solved using substitution?

Yes, the integration of (ln(x))^n can be solved using substitution. This involves choosing a suitable substitution, such as u = ln(x), and then using the substitution rule for integration to solve for the integral.

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