Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.6.50
Chapter 8, Problem 8.6.50

Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 16x^3 (ln(x))^2 dx

Verified step by step guidance
1
Identify the integral to solve: \(\int 16x^{3} (\ln(x))^{2} \, dx\).
Factor out the constant to simplify the integral: \(16 \int x^{3} (\ln(x))^{2} \, dx\).
Use integration by parts, letting \(u = (\ln(x))^{2}\) and \(dv = x^{3} dx\). Then compute \(du\) and \(v\): - \(du = 2 \ln(x) \cdot \frac{1}{x} dx = \frac{2 \ln(x)}{x} dx\) - \(v = \frac{x^{4}}{4}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write \(16 \left( \frac{x^{4}}{4} (\ln(x))^{2} - \int \frac{x^{4}}{4} \cdot \frac{2 \ln(x)}{x} dx \right)\).
Simplify the integral inside and recognize that the new integral is \(\int x^{3} \ln(x) \, dx\), which can be solved using the reduction formula or repeated integration by parts.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Reduction Formulas

Reduction formulas are recursive relations that express an integral involving a power or function in terms of a simpler integral of the same type. They simplify complex integrals by reducing the exponent or power step-by-step, making evaluation manageable through repeated application.
Recommended video:
Guided course
5:59
Recursive Formulas

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It transforms the integral of a product into simpler integrals, often essential when dealing with logarithmic functions multiplied by powers of x.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Properties of Logarithmic Functions in Integration

Logarithmic functions, like ln(x), have unique differentiation and integration properties. Understanding how to handle powers of ln(x) during integration, especially when combined with polynomial terms, is crucial for applying reduction formulas and integration by parts effectively.
Recommended video:
06:21
Properties of Functions
Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to ∞ of (dθ / (θ² - 1))

1
views
Textbook Question

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

1
views
Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₂⁴ dt / [t√(t² − 4)]

Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x² dx) / (4 + x²)

Textbook Question

87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.

Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / √(1 - x²)