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 8.1.40
Chapter 8, Problem 8.1.40

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).

Verified step by step guidance
1
Start by examining the integral: \(\int \frac{\sqrt{x}}{1 + x^{3}} \, dx\). Notice the hint suggests using the substitution \(u = x^{3/2}\), which is equivalent to \(u = x^{\frac{3}{2}}\).
Differentiate \(u\) with respect to \(x\) to find \(du\): Since \(u = x^{3/2}\), then \(\frac{du}{dx} = \frac{3}{2} x^{1/2}\). This implies \(du = \frac{3}{2} x^{1/2} dx\).
Rewrite the integral in terms of \(u\) and \(du\). From the expression for \(du\), solve for \(x^{1/2} dx\): \(x^{1/2} dx = \frac{2}{3} du\). Also, express the denominator \(1 + x^{3}\) in terms of \(u\): since \(u = x^{3/2}\), then \(u^{2} = x^{3}\), so \(1 + x^{3} = 1 + u^{2}\).
Substitute these expressions back into the integral: replace \(\sqrt{x} dx\) with \(\frac{2}{3} du\) and \(1 + x^{3}\) with \(1 + u^{2}\). The integral becomes \(\int \frac{\sqrt{x}}{1 + x^{3}} dx = \int \frac{\frac{2}{3} du}{1 + u^{2}}\).
Now, the integral simplifies to \(\frac{2}{3} \int \frac{1}{1 + u^{2}} du\), which is a standard integral. Recognize that \(\int \frac{1}{1 + u^{2}} du = \arctan(u) + C\). After integrating, substitute back \(u = x^{3/2}\) to express the answer in terms of \(x\).

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.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting u equal a function of x, we rewrite the integral in terms of u and du, making it easier to integrate. This technique is especially useful when the integral contains composite functions.
Recommended video:
07:33
Euler's Method

Algebraic Manipulation of Expressions

Algebraic manipulation involves rewriting expressions to reveal simpler forms or to match substitution patterns. In this problem, expressing √x and x³ in terms of powers of x helps identify a suitable substitution. Recognizing and manipulating exponents is key to simplifying the integral before applying substitution.
Recommended video:
6:36
Simplifying Trig Expressions

Integration of Rational Functions

Integrals involving rational functions, where one polynomial is divided by another, often require substitution or partial fraction decomposition. Understanding how to handle integrals with expressions like 1/(1 + u) after substitution is essential. This knowledge helps in evaluating the integral once the substitution is applied.
Recommended video:
6:04
Intro to Rational Functions
Related Practice
Textbook Question

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

∫₀¹ dr / r^0.999

1
views
Textbook Question

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

1
views
Textbook Question

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(x) dx

Textbook Question

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = c * x * √(25 - x²) over [0, 5]

1
views
Textbook Question

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

1
views
Textbook Question

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].