Textbook Question
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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.
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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.
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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.
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6:04Introduction 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:
07:33Euler's Method
Related Practice
Textbook Question
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
b. ∫ 7x e³ˣ dx
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Textbook Question
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.
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Textbook Question
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?
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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.
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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.
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