Antiderivative of the Function \(\frac{{\ln(x)}}{x}\)

This section is dedicated to finding the antiderivative of the function \(\frac{{\ln(x)}}{x}\), a common problem in calculus.

1. Identify the given function: \(\frac{{\ln(x)}}{x}\), which we want to integrate.

2. Choose a suitable substitution: Let \(u = \ln(x)\). This choice is made because the derivative of \(\ln(x)\) is \(\frac{1}{x}\), which appears in the denominator of our given function.

3. Find the differential \(du\): Since \(u = \ln(x)\), we have \(du = \frac{1}{x} \,dx\).

4. Rewrite the integral in terms of \(u\): Replace \(\ln(x)\) with \(u\) and \(\frac{1}{x} \,dx\) with \(du\) to get \(\int u \,du\).

5. Integrate the expression in terms of \(u\): The antiderivative of \(u\) with respect to \(u\) is \(\frac{u^2}{2}\), so \(\int u \,du = \frac{u^2}{2} + C\), where \(C\) is the constant of integration.

6. Substitute back in terms of \(x\): Replace \(u\) with \(\ln(x)\) to get \(\frac{{(\ln(x))^2}}{2} + C\).

Conclusion: The antiderivative of the function \(\frac{{\ln(x)}}{x}\) is \(\frac{{(\ln(x))^2}}{2} + C\). This detailed analysis of the antiderivative of the function \(\frac{{\ln(x)}}{x}\) provides a comprehensive understanding of the substitution method, a powerful technique in integral calculus.

Antiderivative of the Function \(\frac{{\ln(x)}}{x}\) is an essential concept that illustrates the application of the substitution method in integration. Understanding this example is vital for students and professionals working with calculus.

Antiderivative of the Function \(\frac{{\ln(ax)}}{x}\)

This section is dedicated to finding the antiderivative of the function \(\frac{{\ln(ax)}}{x}\), a variation of the previous example.

1. Identify the given function: \(\frac{{\ln(ax)}}{x}\), which we want to integrate.

2. Choose a suitable substitution: Let \(u = \ln(ax)\). This choice is made because the derivative of \(\ln(ax)\) is \(\frac{1}{x}\), which appears in the denominator of our given function.

3. Find the differential \(du\): Since \(u = \ln(ax)\), we have \(du = \frac{d}{dx}(\ln(ax)) \,dx\). Using the chain rule, the derivative of \(\ln(ax)\) is \(\frac{1}{ax} \cdot a = \frac{1}{x}\), so \(du = \frac{1}{x} \,dx\). Notice that the constant \(a\) cancels out in the derivative.

4. Rewrite the integral in terms of \(u\): Replace \(\ln(ax)\) with \(u\) and \(\frac{1}{x} \,dx\) with \(du\) to get \(\int u \,du\).

5. Integrate the expression in terms of \(u\): The antiderivative of \(u\) with respect to \(u\) is \(\frac{u^2}{2}\), so \(\int u \,du = \frac{u^2}{2} + C\), where \(C\) is the constant of integration.

6. Substitute back in terms of \(x\): Replace \(u\) with \(\ln(ax)\) to get \(\frac{{(\ln(ax))^2}}{2} + C\).

Conclusion: The antiderivative of the function \(\frac{{\ln(ax)}}{x}\) is \(\frac{{(\ln(ax))^2}}{2} + C\). This detailed analysis of the antiderivative of the function \(\frac{{\ln(ax)}}{x}\) provides a comprehensive understanding of the substitution method, a powerful technique in integral calculus.

Antiderivative of the Function \(\frac{{\ln(ax)}}{x}\) is an essential concept that illustrates the application of the substitution method in integration. Understanding this example is vital for students and professionals working with calculus.