How do I show that $F(1/x) = -F(x)$?
Let $F$ be defined by
$F(x) = int_1^x frac{exp(frac{u^2 + 1}{u})du}{u}$.
I'm supposed to show that $F(1/x) = -F(x)$. The hint in the back of my book says that I should use the substitution $v = frac{1}{u}$. When I do this I get the integral
$int_1^x frac{-1}{v^2}exp(frac{1}{v} + v)dv$. Am I supposed to proceed by doing integration by parts?
calculus analysis
add a comment |
Let $F$ be defined by
$F(x) = int_1^x frac{exp(frac{u^2 + 1}{u})du}{u}$.
I'm supposed to show that $F(1/x) = -F(x)$. The hint in the back of my book says that I should use the substitution $v = frac{1}{u}$. When I do this I get the integral
$int_1^x frac{-1}{v^2}exp(frac{1}{v} + v)dv$. Am I supposed to proceed by doing integration by parts?
calculus analysis
1
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37
add a comment |
Let $F$ be defined by
$F(x) = int_1^x frac{exp(frac{u^2 + 1}{u})du}{u}$.
I'm supposed to show that $F(1/x) = -F(x)$. The hint in the back of my book says that I should use the substitution $v = frac{1}{u}$. When I do this I get the integral
$int_1^x frac{-1}{v^2}exp(frac{1}{v} + v)dv$. Am I supposed to proceed by doing integration by parts?
calculus analysis
Let $F$ be defined by
$F(x) = int_1^x frac{exp(frac{u^2 + 1}{u})du}{u}$.
I'm supposed to show that $F(1/x) = -F(x)$. The hint in the back of my book says that I should use the substitution $v = frac{1}{u}$. When I do this I get the integral
$int_1^x frac{-1}{v^2}exp(frac{1}{v} + v)dv$. Am I supposed to proceed by doing integration by parts?
calculus analysis
calculus analysis
edited Nov 29 '18 at 22:39
Bernard
118k639112
118k639112
asked Nov 29 '18 at 22:33
K.MK.M
686312
686312
1
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37
add a comment |
1
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37
1
1
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37
add a comment |
2 Answers
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The definition of $F(x)$ is
$$F(x)=int_1^x e^{u+1/u}frac{du}{u}$$
Substituting $umapsto 1/v$ sends $e^{u+1/u}mapsto e^{v+1/v}$ and $du/umapsto-dv/v$, and replaces your integration bounds with $1$ and $1/x$ respectively, so you have
$$F(x)=int_1^{1/x} e^{v+1/v}cdot -frac{dv}{v}=-F(1/x)$$
and you are done.
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
add a comment |
If you do $v=dfrac1u$ and $mathrm dv=-dfrac{mathrm du}{u^2}$, then your integral becomes$$-int_1^{frac1x}frac{expleft(frac1v+vright)}{frac1v}timesfrac1{v^2},mathrm dv=-int_1^{frac1x}frac{expleft(frac1v+vright)}v,mathrm dv=-Fleft(frac1xright).$$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
The definition of $F(x)$ is
$$F(x)=int_1^x e^{u+1/u}frac{du}{u}$$
Substituting $umapsto 1/v$ sends $e^{u+1/u}mapsto e^{v+1/v}$ and $du/umapsto-dv/v$, and replaces your integration bounds with $1$ and $1/x$ respectively, so you have
$$F(x)=int_1^{1/x} e^{v+1/v}cdot -frac{dv}{v}=-F(1/x)$$
and you are done.
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
add a comment |
The definition of $F(x)$ is
$$F(x)=int_1^x e^{u+1/u}frac{du}{u}$$
Substituting $umapsto 1/v$ sends $e^{u+1/u}mapsto e^{v+1/v}$ and $du/umapsto-dv/v$, and replaces your integration bounds with $1$ and $1/x$ respectively, so you have
$$F(x)=int_1^{1/x} e^{v+1/v}cdot -frac{dv}{v}=-F(1/x)$$
and you are done.
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
add a comment |
The definition of $F(x)$ is
$$F(x)=int_1^x e^{u+1/u}frac{du}{u}$$
Substituting $umapsto 1/v$ sends $e^{u+1/u}mapsto e^{v+1/v}$ and $du/umapsto-dv/v$, and replaces your integration bounds with $1$ and $1/x$ respectively, so you have
$$F(x)=int_1^{1/x} e^{v+1/v}cdot -frac{dv}{v}=-F(1/x)$$
and you are done.
The definition of $F(x)$ is
$$F(x)=int_1^x e^{u+1/u}frac{du}{u}$$
Substituting $umapsto 1/v$ sends $e^{u+1/u}mapsto e^{v+1/v}$ and $du/umapsto-dv/v$, and replaces your integration bounds with $1$ and $1/x$ respectively, so you have
$$F(x)=int_1^{1/x} e^{v+1/v}cdot -frac{dv}{v}=-F(1/x)$$
and you are done.
answered Nov 29 '18 at 22:39
FrpzzdFrpzzd
22.3k839107
22.3k839107
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
add a comment |
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
I just want to make sure that the order of the bounds doesn't matter. So even if $1/x < 1$, we can still write the integral as you've done above?
– K.M
Nov 29 '18 at 23:10
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
@K.M Yes! Recall the following property of integrals: $$int_a^b=-int_b^a$$
– Frpzzd
Nov 29 '18 at 23:48
add a comment |
If you do $v=dfrac1u$ and $mathrm dv=-dfrac{mathrm du}{u^2}$, then your integral becomes$$-int_1^{frac1x}frac{expleft(frac1v+vright)}{frac1v}timesfrac1{v^2},mathrm dv=-int_1^{frac1x}frac{expleft(frac1v+vright)}v,mathrm dv=-Fleft(frac1xright).$$
add a comment |
If you do $v=dfrac1u$ and $mathrm dv=-dfrac{mathrm du}{u^2}$, then your integral becomes$$-int_1^{frac1x}frac{expleft(frac1v+vright)}{frac1v}timesfrac1{v^2},mathrm dv=-int_1^{frac1x}frac{expleft(frac1v+vright)}v,mathrm dv=-Fleft(frac1xright).$$
add a comment |
If you do $v=dfrac1u$ and $mathrm dv=-dfrac{mathrm du}{u^2}$, then your integral becomes$$-int_1^{frac1x}frac{expleft(frac1v+vright)}{frac1v}timesfrac1{v^2},mathrm dv=-int_1^{frac1x}frac{expleft(frac1v+vright)}v,mathrm dv=-Fleft(frac1xright).$$
If you do $v=dfrac1u$ and $mathrm dv=-dfrac{mathrm du}{u^2}$, then your integral becomes$$-int_1^{frac1x}frac{expleft(frac1v+vright)}{frac1v}timesfrac1{v^2},mathrm dv=-int_1^{frac1x}frac{expleft(frac1v+vright)}v,mathrm dv=-Fleft(frac1xright).$$
answered Nov 29 '18 at 22:39
José Carlos SantosJosé Carlos Santos
152k22123225
152k22123225
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add a comment |
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1
You need to change your integration limits after making the substitution.
– Mattos
Nov 29 '18 at 22:37