Evaluate ∫₀¹ log(1 + x) / (1 + x²) dx

To evaluate the integral ∫ from 0 to 1 of log(1 + x) / (1 + x²) dx, we can use integration techniques such as substitution or series expansion. Here’s a step-by-step approach:

Step 1: Series Expansion

We can start by expanding log(1 + x) using its Taylor series around x = 0:

• log(1 + x) = x - x²/2 + x³/3 - x⁴/4 + ...

This series converges for |x| < 1, which is suitable for our limits of integration.

Step 2: Substitute the Series into the Integral

Substituting the series into the integral gives:

∫ from 0 to 1 of (x - x²/2 + x³/3 - x⁴/4 + ...) / (1 + x²) dx.

Step 3: Simplifying the Integral

Now, we can integrate term by term. Each term will be of the form:

• ∫ from 0 to 1 of x^n / (1 + x²) dx.

For each term, we can use the substitution method or integration by parts as needed.

Step 4: Evaluate Each Integral

After evaluating the integrals for each term, we can sum them up. The integral of x^n / (1 + x²) can often be solved using trigonometric substitution or recognizing it as a standard integral.

Final Result

After performing all calculations, the value of the integral ∫ from 0 to 1 of log(1 + x) / (1 + x²) dx is:

This result can be verified through numerical integration or by using advanced calculus techniques if needed.