Integrating Natural Log Function using Integration by Parts Method

In summary, the given problem was to integrate ln(2x+1)dx using the "Integration by Parts" method. The solution involved using the substitution u=ln(2x+1) and applying the formula \int ln{(u)} du = u * (-1 + ln{(u)}) + C. The final result was found to be 0.5(2x+1)ln(2x+1) - x +C.
  • #1
jrmed13
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Integrating Natural Log Function using "Integration by Parts" Method

Homework Statement


The problem says to integrate ln(2x+1)dx


Homework Equations


I used u=ln(2x+1); du = 2dx/(2x+1); dv=dx; v=x


The Attempt at a Solution


So, I integrated it using that (above) 'dictionary' and I got the expression xln(2x+1) - integral of (2x/2x+1)

I could substitute again and say that u=(1/(2x+1)); dv=2xdx, but that process would never end!
And, if I use u=(2x), then various parts of the equation would cancel and I would be left with integral of (ln(2x+1)) = integral of (ln(2x+1))...
I know that the answer should be 0.5(2x+1)ln(2x+1) - x +C, but I can't seem to get it!
 
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  • #2


[tex]\int\ln{(2x+1)}dx=x\ln{(2x+1)}-\int\frac{2x}{2x+1}dx[/tex]

If you add 1 and subtract 1, you can attain your denominator.

[tex]\int\frac{2x+1-1}{2x+1}dx[/tex]

Now break it up, and go from there.
 
  • #3


THANK YOU SO MUCH! (I got the answer :D)
 
  • #4


This is the general rule that can be proved for any type of antideravitve of the the natural log function by use of integration by parts.
[itex]\int ln{(u)} du = u * (-1 + ln{(u)}) + C[/itex]
 

1. What is integration by parts method?

The integration by parts method is a technique used in calculus to find the integral of a product of two functions. It involves breaking down the integral into two parts and using a specific formula to solve for the integral.

2. What is the natural log function?

The natural log function, denoted as ln(x), is the inverse of the exponential function. It is a commonly used function in mathematics and has various applications in fields such as finance, physics, and biology.

3. Why is the integration by parts method used for integrating the natural log function?

The integration by parts method is used for integrating the natural log function because the natural log function is a product of two functions (x and ln(x)), making it a suitable candidate for this method.

4. What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv − ∫v du, where u and v are the two functions being multiplied together, and du and dv represent their respective differentials.

5. What are the steps for integrating the natural log function using the integration by parts method?

The steps for integrating the natural log function using the integration by parts method are as follows:1. Identify u and dv from the given integral.2. Find du and v by taking the derivative and antiderivative of u and dv, respectively.3. Substitute the values of u, du, v, and dv into the integration by parts formula.4. Simplify the resulting integral and solve for the final answer.

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