The integral $$ ext{∫ ln(ax) dx}$$ represents the process of finding the antiderivative of the natural logarithm function multiplied by a linear term. This integral is significant because it combines the properties of logarithmic functions with polynomial expressions, which often appear in various applications, such as physics and economics. Understanding how to evaluate this integral is essential for working with integrals involving exponential and logarithmic functions, especially in integration techniques like integration by parts.
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To evaluate the integral $$ ext{∫ ln(ax) dx}$$, you can use integration by parts, where you let $$u = ext{ln(ax)}$$ and $$dv = dx$$.
The result of integrating $$ ext{ln(ax)}$$ will involve a combination of logarithmic and polynomial expressions, often leading to a simpler form for further calculations.
This integral can be rewritten using properties of logarithms, specifically $$ ext{ln(ax) = ln(a) + ln(x)}$$, simplifying the evaluation process.
Understanding this integral is crucial for solving problems related to area under curves defined by logarithmic functions and their applications.
It's important to remember that when integrating logarithmic functions, you may encounter limits of integration that affect the final answer.
Review Questions
How does integration by parts apply to evaluating the integral $$ ext{∫ ln(ax) dx}$$?
Integration by parts is crucial for evaluating $$ ext{∫ ln(ax) dx}$$ because it allows you to separate the logarithmic function from the differential. By letting $$u = ext{ln(ax)}$$ and $$dv = dx$$, you can find $$du$$ and $$v$$, leading to the formula $$ ext{∫ u dv = uv - ∫ v du}$$. This technique simplifies the integration process and helps derive an expression that is easier to work with.
What simplifications can be made using properties of logarithms when working with the integral $$ ext{∫ ln(ax) dx}$$?
When working with the integral $$ ext{∫ ln(ax) dx}$$, you can simplify it by using the property of logarithms that states $$ ext{ln(ax) = ln(a) + ln(x)}$$. This allows you to break down the integral into two separate parts: one involving a constant (which integrates easily) and one involving $$ ext{ln(x)}$$. This makes evaluating the integral more straightforward and manageable.
Evaluate the integral $$ ext{∫ ln(3x) dx}$$ and explain your reasoning step-by-step.
To evaluate the integral $$ ext{∫ ln(3x) dx}$$, first use integration by parts. Set $$u = ext{ln(3x)}$$ and differentiate it to get $$du = rac{1}{x} dx$$. Next, let $$dv = dx$$ so that $$v = x$$. Apply the integration by parts formula: $$ ext{∫ u dv = uv - ∫ v du}$$. This gives you: $$x ext{ln(3x)} - ext{∫ x * (1/x) dx}$$. The remaining integral simplifies to just $$x$$. So, combining these results yields: $$x ext{ln(3x)} - x + C$$ where C is the constant of integration.