Solving Difficult Integral Homework: \int_0^1\frac{ln(1+x)}{1+x^{2}} dx

• Rats_N_Cats
In summary, the steps for solving this difficult integral include using the substitution u = 1 + x^2, applying integration by parts, and using a calculator if needed. The choice of this substitution makes the integral easier to solve by simplifying the expression and allowing for familiar integration techniques. This technique can also be applied to similar integrals such as \int_0^1\frac{ln(1+x)}{1+x^3}dx.
Rats_N_Cats

Homework Statement

I'm stuck with this definite integral : $$\int_0^1\frac{ln(1+x)}{1+x^{2}} dx$$

Homework Equations

The various "standard integrals".

The Attempt at a Solution

I just don't know where to start, or how to do it. I tried various substitutions but none of them worked; I also tired doing it by parts but that didn't work either.

Last edited:

Okay, I got that. Thanks.
I knew of the identity, but didn't know I'd have to use it after making a substitution.

1. What are the steps for solving this difficult integral?

There are several steps for solving this integral:1. First, use the substitution u = 1 + x^2 to rewrite the integral as \int_1^2\frac{ln(u)}{u}du.2. Next, use integration by parts with u = ln(u) and dv = 1/u to solve the new integral.3. After applying integration by parts, the integral will become \frac{ln(u)^2}{2}\bigg|_1^2 - \int_1^2\frac{ln(u)}{2u}du.4. Use the substitution v = ln(u) to solve the remaining integral.5. Finally, substitute back in the original variable x and evaluate the integral from 0 to 1.

2. Can I use a calculator to solve this integral?

Yes, you can use a calculator to solve this integral. However, it is important to understand the steps and concepts behind the solution rather than relying on a calculator.

3. How does the choice of the substitution u = 1 + x^2 make the integral easier to solve?

The substitution u = 1 + x^2 helps to simplify the integral by getting rid of the quotient and reducing the power of x. It also helps to transform the integral into a familiar form that can be solved using integration techniques such as integration by parts.

4. Is there a specific technique that is most effective for solving this integral?

Yes, the substitution u = 1 + x^2 followed by integration by parts is the most effective technique for solving this integral.

5. Can you provide an example of a similar integral that can be solved using the same techniques?

Yes, an integral of the form \int_0^1\frac{ln(1+x)}{1+x^3}dx can also be solved using the substitution u = 1 + x^3 and integration by parts.

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