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- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.01%3A_Integration_by_PartsThis page provides an overview of integration by parts, a technique used to simplify the integration of products of functions. It includes the formula derived from the product rule, guidance on choosi...This page provides an overview of integration by parts, a technique used to simplify the integration of products of functions. It includes the formula derived from the product rule, guidance on choosing functions with the LIATE mnemonic, and multiple examples ranging from logarithmic to trigonometric integrals. The text also discusses evaluating both definite and indefinite integrals and emphasizes validating results through differentiation.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.03%3A_Trigonometric_SubstitutionThe technique of trigonometric substitution comes in very handy when evaluating integrals of certain forms. This technique uses substitution to rewrite these integrals as trigonometric integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.07%3A_Integrals_Resulting_in_Inverse_Trigonometric_Functions/1.7E%3A_Exercises_for_Section_1.7This page presents exercises on evaluating integrals, emphasizing inverse trigonometric functions and substitutions. It includes definite integrals with solutions, techniques for finding antiderivativ...This page presents exercises on evaluating integrals, emphasizing inverse trigonometric functions and substitutions. It includes definite integrals with solutions, techniques for finding antiderivatives, and addresses undefined integrals. The text also covers integral calculations involving trigonometric and exponential functions, detailing methods to determine constants for definite integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.09%3A_Calculus_of_the_Hyperbolic_FunctionsThis page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives includ...This page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives include understanding derivatives, integrals, and their respective formulas for hyperbolic functions, as well as domain considerations for inverse functions. The text provides examples, exercises, and instructions for evaluating integrals using u-substitution.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.03%3A_Trigonometric_Substitution/3.3E%3A_Exercises_for_Section_3.3This page presents mathematical exercises focused on simplifying trigonometric and hyperbolic expressions, completing the square for trinomials, and performing integrals using trigonometric substituti...This page presents mathematical exercises focused on simplifying trigonometric and hyperbolic expressions, completing the square for trinomials, and performing integrals using trigonometric substitution. It includes notable results like simplifying 9\sec^2θ−9 to 9\tan^2θ and evaluating various integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.05%3A_Other_Strategies_for_Integration/3.5E%3A_Exercises_for_Section_3.5This page presents exercises for evaluating integrals, featuring specific integral problems with solutions, including logarithmic, arctangent, and hyperbolic functions. It includes methods for calcula...This page presents exercises for evaluating integrals, featuring specific integral problems with solutions, including logarithmic, arctangent, and hyperbolic functions. It includes methods for calculating areas and volumes from curves, along with instructions for confirming results using calculators. The document is attributed to OpenStax and encompasses both definite and indefinite integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.05%3A_Other_Strategies_for_IntegrationIn addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which a...In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which are readily available in many books, including the appendices to this one. Also widely available are computer algebra systems (CAS), which are found on calculators and in many campus computer labs, and are free online.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.06%3A_Integrals_Involving_Exponential_and_Logarithmic_Functions/1.6E%3A_Exercises_for_Section_1.6This page covers exercises on computing indefinite and definite integrals using calculus techniques, including substitutions and properties of logarithmic functions. It presents specific integrals to ...This page covers exercises on computing indefinite and definite integrals using calculus techniques, including substitutions and properties of logarithmic functions. It presents specific integrals to solve, discusses the natural logarithm function, its properties, and implications, and provides answers for better understanding.
- https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.06%3A_Triple_Integrals_in_Cylindrical_and_Spherical_Coordinates/2.6E%3A_ExercisesThis page presents exercises focused on evaluating triple integrals over solid regions in three-dimensional space, using cylindrical and spherical coordinates. It includes function transformations, co...This page presents exercises focused on evaluating triple integrals over solid regions in three-dimensional space, using cylindrical and spherical coordinates. It includes function transformations, computations of volumes for shapes like cylinders and cones, and integral evaluations, providing explicit examples and solutions. Additionally, the document emphasizes the conversion between coordinate systems and discusses the properties of continuous functions with symmetry.
- https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_Regions/2.1E%3A_ExercisesThis page covers exercises on estimating volumes and integrals using numerical methods like the midpoint rule and Riemann sums for specific functions over defined regions. It includes discussions on d...This page covers exercises on estimating volumes and integrals using numerical methods like the midpoint rule and Riemann sums for specific functions over defined regions. It includes discussions on double integrals, solid geometry, and inequalities related to these integrals. The text also addresses the average value of functions, specifically calculating the average temperature across a rectangular region, with results and estimations presented in data tables.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.07%3A_Improper_IntegralsIn this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We ...In this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We examine several techniques for evaluating improper integrals, all of which involve taking limits.
