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- https://math.libretexts.org/Bookshelves/Calculus/Applied_Calculus_(Calaway_Hoffman_and_Lippman)/03%3A_The_Integral/3.05%3A_SubstitutionIf the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) ...If the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) and ∫x=bx=a(original integrand)dx becomes ∫u=g(b)u=g(a)(new integrand)du.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2160%3A_Applied_Calculus_I/04%3A_The_Integral/4.06%3A_SubstitutionThe Substitution Method (also called u -Substitution) is one way of algebraically manipulating an integrand so that the rules apply. This is a way to unwind or undo the Chain Rule for derivatives. Wh...The Substitution Method (also called u -Substitution) is one way of algebraically manipulating an integrand so that the rules apply. This is a way to unwind or undo the Chain Rule for derivatives. When you find the derivative of a function using the Chain Rule, you end up with a product of something like the original function times a derivative.
- 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/Penn_State_University_Greater_Allegheny/MATH_110%3A_Techniques_of_Calculus_I_(Gaydos)/03%3A_The_Integral/3.04%3A_SubstitutionThe Substitution Method (also called u -Substitution) is one way of algebraically manipulating an integrand so that the rules apply. This is a way to unwind or undo the Chain Rule for derivatives. Wh...The Substitution Method (also called u -Substitution) is one way of algebraically manipulating an integrand so that the rules apply. This is a way to unwind or undo the Chain Rule for derivatives. When you find the derivative of a function using the Chain Rule, you end up with a product of something like the original function times a derivative.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/08%3A_Techniques_of_Integration/8.02%3A_u-SubstitutionTo summarize: if we suspect that a given function is the derivative of another via the chain rule, we let u denote a likely candidate for the inner function, then translate the given function so t...To summarize: if we suspect that a given function is the derivative of another via the chain rule, we let u denote a likely candidate for the inner function, then translate the given function so that it is written entirely in terms of u, with no x remaining in the expression.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.07%3A_Integrals_Exponential_Functions_and_LogarithmsWe already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponent...We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number e is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.
- https://math.libretexts.org/Courses/Butler_Community_College/MA148%3A_Calculus_with_Applications_-_Butler_CC/03%3A_The_Integral/3.05%3A_SubstitutionIf the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) ...If the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) and ∫x=bx=a(original integrand)dx becomes ∫u=g(b)u=g(a)(new integrand)du.
- https://math.libretexts.org/Courses/Chabot_College/MTH_15%3A_Applied_Calculus_I/05%3A_The_Integral/5.05%3A_SubstitutionIf the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) ...If the original integral had endpoints x=a and x=b, and we make the substitution u=g(x) and du=g′(x)dx, then the new integral will have endpoints u=g(a) and u=g(b) and ∫x=bx=a(original integrand)dx becomes ∫u=g(b)u=g(a)(new integrand)du.
