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Chapter 5: Problem 3

Evaluate the integrals using the indicated substitutions. $$ \begin{array}{l}{\text { (a) } \int \sec ^{2}(4 x+1) d x ; u=4 x+1} \\ {\text { (b) } \int y \sqrt{1+2 y^{2}} d y ; u=1+2 y^{2}}\end{array} $$

Short Answer

Expert verified
(a) \( \frac{1}{4} \tan(4x+1) + C \) (b) \( \frac{1}{6} (1+2y^2)^{3/2} + C \)

Step by step solution

01

Identify the Substitution for Integral (a)

For the integral \( \int \sec^{2}(4x+1) \, dx \), we are given the substitution \( u = 4x+1 \). This means that we need to express \( dx \) in terms of \( du \). First, find the derivative \( \frac{du}{dx} = 4 \). Thus, \( dx = \frac{1}{4} \, du \).
02

Substitute and Integrate for Integral (a)

Substitute \( u = 4x+1 \) and \( dx = \frac{1}{4} \, du \) into the integral: \( \int \sec^{2}(u) \cdot \frac{1}{4} \, du = \frac{1}{4} \int \sec^{2}(u) \, du \). The integral \( \int \sec^{2}(u) \, du \) is known to be \( \tan(u) \). Thus, we have \( \frac{1}{4} \tan(u) + C \).
03

Back Substitute for Integral (a)

Replace \( u \) with the original expression in terms of \( x \): \( u = 4x+1 \). Therefore, the integral becomes \( \frac{1}{4} \tan(4x+1) + C \).
04

Identify the Substitution for Integral (b)

For the integral \( \int y \sqrt{1+2y^2} \, dy \), we are given substitution \( u = 1 + 2y^2 \). This implies we need to express \( dy \) in terms of \( du \). Calculate the derivative \( \frac{du}{dy} = 4y \), hence \( dy = \frac{1}{4y} \, du \).
05

Substitute and Simplify the Integral (b)

Using the substitution \( u = 1 + 2y^2 \) and \( dy = \frac{1}{4y} \, du \), substitute into the integral: \( \int y \sqrt{u} \cdot \frac{1}{4y} \, du = \frac{1}{4} \int \sqrt{u} \, du \). The expression simplifies to \( \frac{1}{4} \int u^{1/2} \, du \).
06

Integrate the Simplified Expression (b)

Integrate \( \frac{1}{4} \int u^{1/2} \, du \) to get \( \frac{1}{4} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{6} u^{3/2} + C \).
07

Back Substitute for Integral (b)

Replace \( u \) with \( 1 + 2y^2 \) to get the final expression of the integral: \( \frac{1}{6} (1 + 2y^2)^{3/2} + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful technique for solving integrals, especially when dealing with complex expressions. It simplifies the process by transforming the integral into a more workable form. In essence, you replace a complicated part of the integrand with a single variable.
Here's how the substitution method typically works:
  • Select a substitution: Choose a part of the integral expression to replace with a new variable. This is often a natural choice if it's a more intricate expression, like the one given in the tasks.
  • Derive the new variable: Find the derivative of your substitution with respect to the original variable. This helps in expressing 'dx' or 'dy' in terms of 'du'.
  • Transform the integral: Substitute both the original variable and its derivative into the integral, changing it completely into the new variable.
  • Integrate: Now, integrate with respect to the new variable, which is often simpler.
  • Back-substitute: Finally, replace the new variable with the original expression to return to the terms of the original variable.
This technique is particularly useful as it changes a difficult integral into a form that can be dealt with using basic integration rules, as seen in problems (a) and (b) above.
Indefinite Integrals
Indefinite integrals are fundamental to calculus as they represent a class of functions rather than a single numerical value, as with definite integrals. An indefinite integral is generally expressed as \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is the constant of integration.
Some key points about indefinite integrals include:
  • Antiderivatives: Finding an indefinite integral involves determining the antiderivative of a function. This is essentially the reverse process of differentiation.
  • Constant of Integration \(C\): The "\(+ C\)" represents an arbitrary constant, signifying that there are infinitely many antiderivatives, each differing by a constant.
  • Techniques: Various techniques, such as substitution and integration by parts, help solve indefinite integrals, adapting to the complexity of the function.
When you apply these principles to examples such as \( \int \sec^{2}(4x+1) \, dx \) or \( \int y \sqrt{1+2y^{2}} \, dy \), you harness specific integration rules and substitution to find these antiderivatives, finally verifying them by differentiation.
Trigonometric Integrals
Trigonometric integrals involve integrating functions of trigonometric expressions such as \( \sin x \), \( \cos x \), and \( \sec x \). They are quite common and often require strategic techniques due to their cyclical nature.
Here are some strategies used for integrating trigonometric functions:
  • Known Integrals: Many trigonometric integrals are already well-documented, such as \( \int \sec^{2} x \, dx = \tan x + C \), simplifying integration if they match a standard form.
  • Substitution: Sometimes a simple substitution, as shown in the example above, can simplify an otherwise complex trigonometric integral into a basic form.
  • Trigonometric Identities: Using identities such as \( \sin^2 x + \cos^2 x = 1 \) or the double angle formulas can transform the integrals into manageable forms.
  • Trigonometric Substitution: This involves using trigonometric identities to substitute and transform non-trigonometric integrals, often used when dealing with square root terms.
By carefully selecting and applying these methods, as seen in the initial problem set, trigonometric integrals can be simplified and integrated successfully, letting you find expressions for antiderivatives or other analytical forms quickly.

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Most popular questions from this chapter

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