Understanding the Integration of dx/(x(1 log x^2)) from 1 to e^x

Integrals involving logarithmic functions, such as dx/(x(1 log x^2)), are common in advanced calculus. Understanding and solving such integrals can be challenging but is fundamental for students and professionals dealing with various scientific and engineering applications.

Conceptual Overview

The integral in question, ∫ dx/(x(1 log x^2)), with the upper limit e^x and lower limit 1, is a classic example that requires a strategic substitution to simplify the problem. The concept of substitution in integration is a powerful tool that helps transform complex integrals into simpler, more manageable forms.

Substitution Method

One of the most effective ways to integrate such expressions is by using a suitable substitution. In this case, the substitution 1 log x^2 t can significantly simplify the integral. Let's explore why and how this works.

Step-by-Step Solution

Step 1: Define the Substitution

Let 1 log x^2 t. This substitution can be done for several reasons:

It simplifies the denominator of the integrand. It allows us to convert the integral into a form that is easier to solve. It helps in finding the differential relationship between x and t.

Step 2: Compute the Differential dt

First, differentiate both sides of the substitution:

1 log x^2 t

Partial differentiation with respect to x:

0 (1/x) * 2x (dt/dx)

Simplify to get:

2/x dt/dx

Thus, we have:

dx (x/2) * dt

Substituting this into the original integral:

∫ dx/(x(1 log x^2)) ∫ (x/2) * dt / (x(1 log x^2))

Since 1 log x^2 t, the integral simplifies to:

∫ (1/2) * dt / t

Step 3: Integrate with Respect to t

The integral now becomes:

(1/2) * ∫ dt / t

Integration of 1/t is a standard integral, giving:

(1/2) * ln|t| C

Step 4: Back-Substitute t 1 log x^2

Substitute t back into the integral:

(1/2) * ln|1 log x^2| C

Evaluating the Definite Integral

To find the value of the definite integral from 1 to e^x, we need to evaluate the expression at the upper and lower limits and subtract the results:

_integral (1/2) * ln|1 log x^2| at upper limit - (1/2) * ln|1 log x^2| at lower limit

At the upper limit x e^x:

1 log (e^x)^2 1 2x

Thus, at x e^x:

(1/2) * ln|1 2e^x|

At the lower limit x 1:

1 log 1^2 1 0 1

Thus, at x 1:

(1/2) * ln|1| 0

Subtracting the values:

(1/2) * ln(1 2e^x) - 0

The final result is:

(1/2) * ln(1 2e^x)

Conclusion

The integral ∫ dx/(x(1 log x^2)) from 1 to e^x simplifies to (1/2) * ln(1 2e^x). Understanding this process and the substitution method is crucial for solving similar integrals that involve logarithmic functions.

References

1. Calculus: Early Transcendentals, by James Stewart.

2. Integral Calculus, by M.L. Khanna.

3. Techniques of Integration, by James Nicholson.