Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.68b
Chapter 8, Problem 8.1.68b

68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.

Verified step by step guidance
1
Step 1: Begin by identifying the substitution u = csc(x). This substitution simplifies the integral by expressing cot(x) and csc²(x) in terms of u.
Step 2: Compute the derivative of u with respect to x. Since u = csc(x), we know that du/dx = -csc(x)cot(x). Rearrange this to express dx in terms of du: dx = -du / (csc(x)cot(x)).
Step 3: Rewrite the integral ∫(cot(x)csc²(x))dx using the substitution. Replace csc(x) with u and dx with -du / (csc(x)cot(x)). This transforms the integral into ∫(cot(x)u²)(-du / (u cot(x))).
Step 4: Simplify the expression. Notice that cot(x) cancels out, leaving ∫(-u²)du. The integral is now in terms of u, which is easier to evaluate.
Step 5: Integrate -u² with respect to u. The result will be in terms of u, and you can substitute back u = csc(x) to express the final 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:
2m
Was this helpful?

Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, substitution is particularly useful as it simplifies the integral by changing variables, making it easier to evaluate.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as cotangent (cot) and cosecant (csc), are fundamental in calculus, especially in integrals involving angles. Understanding their relationships and identities is crucial for manipulating and simplifying expressions during integration. For instance, csc²(x) is the derivative of -cot(x), which can be leveraged in the integration process.
Recommended video:
6:04
Introduction to Trigonometric Functions

Substitution Method

The substitution method is a technique in calculus where a new variable is introduced to simplify the integration process. By letting u = csc(x), the integral can be transformed into a more manageable form. This method is particularly effective when the integrand contains a function and its derivative, allowing for easier evaluation of the integral.
Recommended video:
07:33
Euler's Method
Related Practice
Textbook Question

76. Different Substitutions

b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

Textbook Question

88. Incorrect Calculation

b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.

1
views
Textbook Question

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

b. Is L concave up or concave down on [0, ∞)?

1
views
Textbook Question

{Use of Tech} Powers of sine and cosine It can be shown that

∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =

{

[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even

[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd

}

b. Evaluate the integrals with n = 10 and confirm the result.

Textbook Question

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.

1
views
Textbook Question

Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).


b. Find the volume of the solid generated when R is revolved about the x-axis.


1
views