Problem 52 - Essential Calculus

Question

Find the indefinite integral. $$ \int \ln \left(e^{2 x-1}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \ln \left(e^{2x-1}\right) dx\) is \(x^2 - x + C\).

Step 1: Simplify Logarithmic Expression

Start by using the property of logarithms that \(\ln a^b = b \ln a\), to simplify the function inside the integral: \[\int \ln \left(e^{2x-1}\right) dx = \int (2x-1) \ln (e) dx\] Since \(\ln (e) = 1\), the integrand simplifies to \((2x - 1)\). Thus the integral is: \[\int (2x - 1) dx \]

Step 2: Apply Power Rule for Integration

Apply the power rule for integration which states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). The rule is applied to both terms in the integrand separately. The power rule is applied to \(2x\) treating \(2x\) as \(2*x^1\) and to '-1' treating it as \(-1*x^0\). \[ \int (2x - 1) dx = \int 2x dx - \int dx = x^2 - x \]

Step 3: Add Constant of Integration

Finally, when finding an indefinite integral, always add a constant of integration, typically denoted as 'C'. Thus, the solution to the integral is: \[x^2 - x + C\]

Key Concepts

Logarithmic Expression

In calculus, a logarithmic expression often involves the natural logarithm, denoted as \( \ln \). When a logarithmic function is involved in an integral, it must sometimes be simplified using rules of logarithms. One such important property is \( \ln a^b = b \ln a \). This transformation helps reduce the complexity of the integral by expanding or simplifying the logarithm.

• This property is handy because it can turn products or powers within a logarithm into a more manageable form.
• In the given example, \( \ln(e^{2x-1}) \) leverages this rule.
• The exponent \( 2x - 1 \) is brought down in front of the \( \ln \) function, simplifying the expression to \( (2x - 1) \ln(e) \).

Since \( \ln(e) = 1 \), the expression \( (2x - 1) \ln(e) \) reduces further to just \( 2x - 1 \).
This simplification is crucial because it transforms a potentially complex logarithmic integral into a straightforward polynomial one.

Power Rule for Integration

The power rule for integration is a fundamental technique used when dealing with polynomials. It states that the integral of \( x^n \), where \( n eq -1 \), is \( \frac{x^{n+1}}{n+1} \) plus a constant.

• The power rule is applied to each term of the polynomial within the integral separately.
• In the problem above, the expression \( 2x - 1 \) is split into two terms: \( 2x \) and \( -1 \).
• For \( 2x \), treat it as \( 2x^1 \), applying the rule results in \( \int 2x \, dx = x^2 \).
• For \( -1 \), or \( -1x^0 \), it simplifies to \( \int -1 \, dx = -x \).

After applying the rule to each term, combine them to achieve the result \( x^2 - x \).
This method of integrating polynomials is straightforward and efficient, making it an essential tool in the integration toolkit.

Constant of Integration

Whenever you compute an indefinite integral, it's customary to add a constant of integration, denoted by \( C \). This constant is crucial because it represents the family of all antiderivatives derived from the integrated function.

• The indefinite integral gives a general solution, not a specific one.
• \( C \) adjusts for any vertical shifts in the antiderivative's graph.

In the example provided, after integrating \( 2x - 1 \), the result is \( x^2 - x \). Therefore, including the constant of integration, the complete solution is \( x^2 - x + C \).
This constant ensures the solution encompasses every possible antiderivative of a function, acknowledging that differentiation of any constant yields zero. Hence, the practical utility of \( C \) lies in solving real-world problems where initial conditions or specific solutions are necessary.