Integration of log(x) under √(x^2 + a^2)

To integrate the function log(x) under √(x^2 + a^2), we can use the technique of substitution. Let's break down the steps involved in solving this integral.

1. Substitution:
We begin by making a substitution to simplify the integral. Let's substitute x = a * tanh(u), where u is a new variable. This substitution is chosen because it simplifies the expression √(x^2 + a^2).

2. Calculating differentials:
We need to calculate the differentials dx and du in terms of du. Using the substitution x = a * tanh(u), we can differentiate both sides of this equation with respect to u to obtain:
dx = a * sech^2(u) * du.

3. Substituting differentials:
Now, let's substitute the differentials dx and du back into the integral:
∫log(x) * dx = ∫log(a * tanh(u)) * (a * sech^2(u)) * du.

4. Simplifying the expression:
By using logarithmic properties, we can rewrite the integral as:
∫log(a * tanh(u)) * (a * sech^2(u)) * du = a * ∫(log(a) + log(tanh(u))) * sech^2(u) * du.

5. Distributing the integral:
We can distribute the integral and rewrite it as:
a * (∫log(a) * sech^2(u) * du + ∫log(tanh(u)) * sech^2(u) * du).

6. Evaluating the two integrals:
We can evaluate the two integrals separately. The integral of log(a) * sech^2(u) * du can be simplified to log(a) * tanh(u). To integrate log(tanh(u)) * sech^2(u) * du, we can use integration by parts.

7. Final integration:
After evaluating the integrals, we obtain:
a * (log(a) * tanh(u) + u - log(tanh(u))).

8. Back substitution:
Finally, we substitute back u = tanh^(-1)(x/a) into the expression we obtained in step 7. This gives us the final result of the integral.

In conclusion, by using the technique of substitution and simplifying the expression, we can integrate log(x) under √(x^2 + a^2). The final result will involve logarithmic and inverse hyperbolic functions.