Problem 14 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{1}{t\left[1+(\ln t)^{2}\right]} d t $$

Step-by-Step Solution

Answer

The integral of \(\int \frac{1}{t\left[1+(\ln t)^{2}\right]} dt\) is \( \ln|t| [1+(\ln t)^{2}] + C \)

Step 1: Identify the form

The integral can be compared with the derivative of \(ln|x|\) with respect to x which is \(\frac{1}{x}\). Adding \(1\) to \((\ln t)^{2}\) wouldn't affect the derivative. Hence, rewrite the integral as: \(\int \frac{1}{t} \left[1+(\ln t)^{2}\right] dt \)

Step 2: Apply the integral

Now, using the integral tables, \(\int \frac{1}{t} dt \) is \( \ln|t| + C \) where C is the constant of integration. Hence, the integral of the function can be determined as: \(\int \frac{1}{t} \left[1+(\ln t)^{2}\right] dt = \ln|t| + C \)

Step 3: Final Result

The final result is a function of \(t\). Thus, simplifying the equation, get: \( \ln|t| [1+(\ln t)^{2}] + C \)