Proof of exponential function integral
I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule
$$int a^x dx= frac{a^x}{ln(a)}+C$$
would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.
calculus integration proof-writing exponential-function
asked Dec 16 at 23:16
Math Bob
138
add a comment |
I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule
$$int a^x dx= frac{a^x}{ln(a)}+C$$
would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.
calculus integration proof-writing exponential-function
asked Dec 16 at 23:16
Math Bob
138
5
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46
add a comment |
I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule
$$int a^x dx= frac{a^x}{ln(a)}+C$$
would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.
calculus integration proof-writing exponential-function
asked Dec 16 at 23:16
Math Bob
138
I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule
$$int a^x dx= frac{a^x}{ln(a)}+C$$
would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.
calculus integration proof-writing exponential-function
calculus integration proof-writing exponential-function
asked Dec 16 at 23:16
Math Bob
138
asked Dec 16 at 23:16
Math Bob
138
edited Dec 16 at 23:54
asked Dec 16 at 23:16
Math Bob
138
asked Dec 16 at 23:16
Math Bob
138
asked Dec 16 at 23:16
Math Bob
138
138
5
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46
add a comment |
5
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46
5
5
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46
add a comment |
3 Answers
3
active
oldest
votes
I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,
$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$
If so, then note:
$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$
This is mostly just manipulation of various logarithm properties: namely,
$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$
Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.
answered Dec 16 at 23:23
Eevee Trainer
3,370225
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
add a comment |
Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way
$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$
Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to
$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$
From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to
$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$
$$int a^xdx=frac{a^x}{ln(a)}+k$$
answered Dec 16 at 23:25
mrtaurho
3,0901930
add a comment |
Just differentiate the right hand side and see what you get. Note that
$$
frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
$$
where we used the chain rule in the second equality.
answered Dec 16 at 23:44
Foobaz John
20.7k41250
add a comment |
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3 Answers
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3 Answers
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I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,
$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$
If so, then note:
$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$
This is mostly just manipulation of various logarithm properties: namely,
$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$
Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.
answered Dec 16 at 23:23
Eevee Trainer
3,370225
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
add a comment |
I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,
$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$
If so, then note:
$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$
This is mostly just manipulation of various logarithm properties: namely,
$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$
Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.
answered Dec 16 at 23:23
Eevee Trainer
3,370225
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
add a comment |
I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,
$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$
If so, then note:
$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$
This is mostly just manipulation of various logarithm properties: namely,
$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$
Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.
answered Dec 16 at 23:23
Eevee Trainer
3,370225
I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,
$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$
If so, then note:
$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$
This is mostly just manipulation of various logarithm properties: namely,
$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$
Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.
answered Dec 16 at 23:23
Eevee Trainer
3,370225
answered Dec 16 at 23:23
Eevee Trainer
3,370225
answered Dec 16 at 23:23
Eevee Trainer
3,370225
answered Dec 16 at 23:23
Eevee Trainer
3,370225
3,370225
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
add a comment |
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
– Math Bob
Dec 16 at 23:51
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
– Eevee Trainer
Dec 17 at 0:18
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
Okay. I have taken this in so thanks for the answer!
– Math Bob
Dec 17 at 4:04
add a comment |
Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way
$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$
Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to
$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$
From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to
$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$
$$int a^xdx=frac{a^x}{ln(a)}+k$$
answered Dec 16 at 23:25
mrtaurho
3,0901930
add a comment |
Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way
$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$
Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to
$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$
From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to
$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$
$$int a^xdx=frac{a^x}{ln(a)}+k$$
answered Dec 16 at 23:25
mrtaurho
3,0901930
add a comment |
Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way
$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$
Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to
$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$
From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to
$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$
$$int a^xdx=frac{a^x}{ln(a)}+k$$
answered Dec 16 at 23:25
mrtaurho
3,0901930
Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way
$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$
Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to
$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$
From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to
$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$
$$int a^xdx=frac{a^x}{ln(a)}+k$$
answered Dec 16 at 23:25
mrtaurho
3,0901930
answered Dec 16 at 23:25
mrtaurho
3,0901930
answered Dec 16 at 23:25
mrtaurho
3,0901930
answered Dec 16 at 23:25
mrtaurho
3,0901930
3,0901930
add a comment |
add a comment |
Just differentiate the right hand side and see what you get. Note that
$$
frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
$$
where we used the chain rule in the second equality.
answered Dec 16 at 23:44
Foobaz John
20.7k41250
add a comment |
Just differentiate the right hand side and see what you get. Note that
$$
frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
$$
where we used the chain rule in the second equality.
answered Dec 16 at 23:44
Foobaz John
20.7k41250
add a comment |
Just differentiate the right hand side and see what you get. Note that
$$
frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
$$
where we used the chain rule in the second equality.
answered Dec 16 at 23:44
Foobaz John
20.7k41250
Just differentiate the right hand side and see what you get. Note that
$$
frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
$$
where we used the chain rule in the second equality.
answered Dec 16 at 23:44
Foobaz John
20.7k41250
answered Dec 16 at 23:44
Foobaz John
20.7k41250
answered Dec 16 at 23:44
Foobaz John
20.7k41250
answered Dec 16 at 23:44
Foobaz John
20.7k41250
20.7k41250
add a comment |
add a comment |
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5
It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
Dec 16 at 23:21
@Jakobian : The only real answer. +1.
– MPW
Dec 16 at 23:46