Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter covers a set of essential techniques for evaluating both indefinite and definite integrals. It introduces integration by substitution and integration by parts as two fundamental methods for finding antiderivatives. The chapter also explores the evaluation of definite integrals, techniques for numerical approximation when a closed form is not obtainable, and an exploration of improper integrals. Understanding these techniques is crucial for solving a wide range of problems in mathematics and applied sciences.

Learning Objectives

1

Apply integration techniques such as substitution and integration by parts to evaluate integrals.

2

Evaluate definite integrals using proper limit adjustment and fundamental theorem of calculus.

3

Approximate definite integrals numerically when an analytic antiderivative is not available.

4

Analyze and solve improper integrals by understanding convergence criteria.

5

Demonstrate real-world applications of integration techniques in physical and engineering problems.

Key Concepts

CONCEPT

DEFINITION

Integration by Substitution

A technique that simplifies an integral by making a substitution of variables, effectively reversing the chain rule from differentiation.

Integration by Parts

A method based on the product rule for differentiation; it transforms the integral of a product of functions into an expression involving the integrals of their derivatives.

Definite Integral

An integral evaluated over a specific interval [a, b] that represents the net area under a curve; it is computed using the limits of integration.

Numerical Approximation of Integrals

Techniques for estimating the value of a definite integral using methods such as the Trapezoidal Rule or Simpson's Rule, especially when an antiderivative is difficult to find.

Improper Integral

An integral where either the interval of integration is infinite or the integrand becomes infinite within the interval; these are evaluated as limits to determine convergence or divergence.

Applications of Integration

The use of integrals in solving practical problems in physics, engineering, economics, and other fields by determining areas, volumes, and accumulated quantities.

Example Problems

Example 1

Determine the integrals by making appropriate substitutions. $$\int 2 x\left(x^{2}+4\right)^{5} d x$$

Example 2

Determine the integrals by making appropriate substitutions. $$\int 2(2 x-1)^{7} d x$$

Example 3

Determine the integrals by making appropriate substitutions. $$\int \frac{2 x+1}{\sqrt{x^{2}+x+3}} d x$$

Example 4

Determine the integrals by making appropriate substitutions. $$\int\left(x^{2}+2 x+3\right)^{6}(x+1) d x$$

Example 5

Determine the integrals by making appropriate substitutions. $$\int 3 x^{2} e^{\left(x^{3}-1\right)} d x$$
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Step-by-Step Explanations

QUESTION

Evaluate the integral ∫2x cos(x²) dx using integration by substitution.

STEP-BY-STEP ANSWER:

Step 1: Let u = x², then differentiate: du = 2x dx.
Step 2: Substitute u and du into the integral so that ∫2x cos(x²) dx becomes ∫cos(u) du.
Step 3: Integrate cos(u) with respect to u to obtain sin(u) + C.
Step 4: Replace u with x² to get the final answer: sin(x²) + C.
Final Answer: sin(x²) + C

Integration by Substitution

QUESTION

Evaluate the integral ∫x e^x dx using integration by parts.

STEP-BY-STEP ANSWER:

Step 1: Choose u = x (so that du = dx) and dv = e^x dx (so that v = e^x).
Step 2: Apply the integration by parts formula: ∫u dv = uv - ∫v du.
Step 3: Substitute the chosen parts: ∫x e^x dx = x e^x - ∫e^x dx.
Step 4: Integrate e^x dx to get e^x, leading to: x e^x - e^x + C.
Final Answer: e^x (x - 1) + C

Integration by Parts

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Common Mistakes

  • Failing to properly change the limits of integration when performing substitution in a definite integral.
  • Mixing up the roles of u and du, which can lead to incorrect substitutions.
  • Overlooking the need to adjust terms or signs when applying the integration by parts formula.
  • Assuming every integral has an antiderivative expressible in elementary functions.
  • Neglecting the convergence criteria for improper integrals and incorrectly determining whether they converge or diverge.