When integrating, what are all the exact requirements for a substitution to be valid?
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There are many examples on this site of substitutions that are not valid for varying reasons. But there are probably many more that I still don't know of.
When working with definite and indefinite integrals, what are all the rules for valid substitutions?
(It would also be nice to have explanations but this is not necessary.)
calculus integration
asked Nov 21 at 11:56
PKBeam
3129
add a comment |
up vote
1
down vote
favorite
There are many examples on this site of substitutions that are not valid for varying reasons. But there are probably many more that I still don't know of.
When working with definite and indefinite integrals, what are all the rules for valid substitutions?
(It would also be nice to have explanations but this is not necessary.)
calculus integration
asked Nov 21 at 11:56
PKBeam
3129
I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
There are many examples on this site of substitutions that are not valid for varying reasons. But there are probably many more that I still don't know of.
When working with definite and indefinite integrals, what are all the rules for valid substitutions?
(It would also be nice to have explanations but this is not necessary.)
calculus integration
asked Nov 21 at 11:56
PKBeam
3129
There are many examples on this site of substitutions that are not valid for varying reasons. But there are probably many more that I still don't know of.
When working with definite and indefinite integrals, what are all the rules for valid substitutions?
(It would also be nice to have explanations but this is not necessary.)
calculus integration
calculus integration
asked Nov 21 at 11:56
PKBeam
3129
asked Nov 21 at 11:56
PKBeam
3129
asked Nov 21 at 11:56
PKBeam
3129
asked Nov 21 at 11:56
PKBeam
3129
asked Nov 21 at 11:56
PKBeam
3129
3129
I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36
add a comment |
I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36
I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36
add a comment |
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I think all monotone derivable functions $f$ which derivate doesn't vanish.
– P De Donato
Nov 21 at 12:18
It's best to remember the rule of substitution. There are some variations of the rule and hence it may be described differently in various textbooks. Follow the one prescribed in your textbook. Apostol gives it as: $int_{g(a)} ^{g(b)} f(x) , dx=int_{a}^{b} f(g(t)) g'(t) , dt$ if $g'$ is continuous on $[a, b] $ and $f$ is continuous on $g([a, b]) $.
– Paramanand Singh
Nov 21 at 14:36