how do you integrate ln(x^2+1)dx?
You will find the answer and the method of solution (integration by parts) at
http://answers.yahoo.com/question/index?qid=20080112133411AAk40X1
thank you. i understand now.
To integrate ln(x^2 + 1)dx, we can use a technique called substitution. Here's how you can do it step by step:
Step 1: Identify a suitable substitution. Let's substitute u = x^2 + 1. This choice is based on the derivative of u to be present in the integral.
Step 2: Find du/dx, and solve it for dx:
Differentiating both sides with respect to x, we get: 2x dx = du.
Solving for dx, we have: dx = du / (2x).
Step 3: Rewrite the integral in terms of u:
Substituting u = x^2 + 1 and dx = du / (2x) in the original integral, we have:
∫ ln(u) du / (2x)
Step 4: Simplify the integral:
To simplify further, we can rewrite x in terms of u using the substitution u = x^2 + 1. Since x = √(u - 1), we have:
∫ ln(u) du / (2√(u - 1))
Step 5: Integrate the simplified integral:
Now, we need to integrate the expression ∫ ln(u) du / (2√(u - 1)). This can be done using standard integration techniques for logarithmic functions.
The final result would be:
∫ ln(x^2 + 1) dx = ∫ ln(u) du / (2√(u - 1))
Integration of ln(u) can be done by parts or other methods depending on the level of complexity. It is recommended to use a table of integrals or suitable software to obtain the exact form of the integral.