Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.PE.52
Chapter 7, Problem 7.PE.52

Evaluate the integrals in Exercises 31–78.
52. ∫(from 1 to 32)(1/5x) dx

Verified step by step guidance
1
Identify the integral to be evaluated: \(\int_{1}^{32} \frac{1}{5x} \, dx\).
Rewrite the integrand by factoring out constants: \(\frac{1}{5} \int_{1}^{32} \frac{1}{x} \, dx\).
Recall the integral formula for \(\int \frac{1}{x} \, dx = \ln|x| + C\).
Apply the definite integral evaluation: \(\frac{1}{5} \left[ \ln|x| \right]_{1}^{32} = \frac{1}{5} (\ln 32 - \ln 1)\).
Simplify the expression using properties of logarithms, noting that \(\ln 1 = 0\), so the result is \(\frac{1}{5} \ln 32\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Definite Integral

A definite integral calculates the net area under a curve between two specific points on the x-axis. It is represented as ∫ from a to b of f(x) dx, where a and b are the limits of integration. The result is a number representing the accumulated value of the function over that interval.
Recommended video:
05:43
Definition of the Definite Integral

Integral of a Power Function

The integral of x raised to a power n (where n ≠ -1) is found using the formula ∫ x^n dx = (x^(n+1)) / (n+1) + C. This rule helps in integrating functions like x raised to fractional or negative powers, which is essential for solving integrals involving terms like x^(-1).
Recommended video:
07:32
Representing Functions as Power Series

Properties of Constants in Integration

When integrating, constants can be factored out of the integral to simplify calculations. For example, ∫ c * f(x) dx = c * ∫ f(x) dx, where c is a constant. This property allows easier handling of integrals with constant coefficients, such as 1/5 in the given integral.
Recommended video:
06:21
Properties of Functions
Related Practice
Textbook Question

Evaluate the integrals in Exercises 31–78.

77. ∫dt/((t+1)√(t²+2t-8))

Textbook Question

23. What roles do the functions e^x and ln(x) play in growth comparisons?

Textbook Question

Verify the integration formulas in Exercises 37–40.

37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

1
views
Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

69. ∫(from 5/4 to 2)dx/(1-x²)

Textbook Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

17. y = ln(arccos(x))

Textbook Question

Evaluate the integrals in Exercises 31–78.

37. ∫(from -1 to 1)dx/(3x-4)

1
views