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Chapter 6: Problem 14

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

Short Answer

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

Step by step solution

01

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: \(\int \frac{1}{t} \left[1+(\ln t)^{2}\right] dt \)
02

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: \(\int \frac{1}{t} \left[1+(\ln t)^{2}\right] dt = \ln|t| + C \)
03

Final Result

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

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