143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.

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Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).

Choose \(u = \ln(x)\) and \(dv = dx\). Then, compute \(du = \frac{1}{x} dx\) and \(v = x\).

Apply the integration by parts formula: \(\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx\).

Simplify the integral: \(\int x \cdot \frac{1}{x} \, dx = \int 1 \, dx = x\).

Combine the results to write the integral as \(x \ln(x) - x + C\), where \(C\) is the constant of integration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv − ∫v du. Choosing appropriate u and dv simplifies the integral, as in integrating ln(x) by setting u = ln(x) and dv = dx.

Properties of the Natural Logarithm Function

The natural logarithm function, ln(x), is defined for x > 0 and has the derivative d/dx [ln(x)] = 1/x. Understanding its domain and derivative is essential when integrating ln(x), as it guides the choice of functions in integration by parts and ensures the integral is correctly evaluated.

Constant of Integration

When computing indefinite integrals, the result includes an arbitrary constant C because differentiation of a constant is zero. This constant accounts for all possible antiderivatives, ensuring the general solution to the integral is complete.

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Textbook Question

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_2(x²)

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Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

3. a. arcsin(-1/2)