Question

Evaluate the integral: \[ \int \frac{\log(1+x)}{1+x} \, dx \]

Hint:

When you encounter integrals involving logarithmic functions, try substitution to simplify the integrand.

Answer 1

We can use substitution to solve this integral. Let: \[ u = \log(1+x), \quad du = \frac{1}{1+x} \, dx \] This transforms the integral into: \[ \int u \, du = \frac{1}{2} u^2 + C \] Substituting back for \(u\): \[ \frac{1}{2} \log^2(1+x) + C \]