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Antiderivative of $3cos(x^3)sinx$
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I am practicing for an exam by randomly picking an expression to anti-differentiate when I came up with this one: $3cos(x^3)sinx$
. How would I go about tackling this? I tried using integration by parts:
$$int3cos(x^3)sinxdx = 3cos(x^3)(-cosx)-intsin(x)*-9x^2sin(x^3) dx$$
This just ends up leaving the integral on the RHS even more complicated, in my opinion. I would appreciate it if somebody could show me how to solve this antiderivative.
Thanks in advance.
calculus integration
asked Jul 19 at 23:20
RayDansh
884214
 |Â
show 4 more comments
up vote
0
down vote
favorite
I am practicing for an exam by randomly picking an expression to anti-differentiate when I came up with this one: $3cos(x^3)sinx$
. How would I go about tackling this? I tried using integration by parts:
$$int3cos(x^3)sinxdx = 3cos(x^3)(-cosx)-intsin(x)*-9x^2sin(x^3) dx$$
This just ends up leaving the integral on the RHS even more complicated, in my opinion. I would appreciate it if somebody could show me how to solve this antiderivative.
Thanks in advance.
calculus integration
asked Jul 19 at 23:20
RayDansh
884214
4
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
3
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
1
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
1
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am practicing for an exam by randomly picking an expression to anti-differentiate when I came up with this one: $3cos(x^3)sinx$
. How would I go about tackling this? I tried using integration by parts:
$$int3cos(x^3)sinxdx = 3cos(x^3)(-cosx)-intsin(x)*-9x^2sin(x^3) dx$$
This just ends up leaving the integral on the RHS even more complicated, in my opinion. I would appreciate it if somebody could show me how to solve this antiderivative.
Thanks in advance.
calculus integration
asked Jul 19 at 23:20
RayDansh
884214
I am practicing for an exam by randomly picking an expression to anti-differentiate when I came up with this one: $3cos(x^3)sinx$
. How would I go about tackling this? I tried using integration by parts:
$$int3cos(x^3)sinxdx = 3cos(x^3)(-cosx)-intsin(x)*-9x^2sin(x^3) dx$$
This just ends up leaving the integral on the RHS even more complicated, in my opinion. I would appreciate it if somebody could show me how to solve this antiderivative.
Thanks in advance.
calculus integration
asked Jul 19 at 23:20
RayDansh
884214
asked Jul 19 at 23:20
RayDansh
884214
asked Jul 19 at 23:20
RayDansh
884214
asked Jul 19 at 23:20
RayDansh
884214
884214
4
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
3
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
1
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
1
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05
 |Â
show 4 more comments
4
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
3
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
1
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
1
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05
4
4
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
3
3
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
1
1
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
1
1
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05
 |Â
show 4 more comments
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^x^2$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.
A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.
answered Jul 21 at 17:12
Imari1234
562
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^x^2$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.
A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.
answered Jul 21 at 17:12
Imari1234
562
add a comment |Â
up vote
4
down vote
accepted
It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^x^2$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.
A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.
answered Jul 21 at 17:12
Imari1234
562
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^x^2$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.
A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.
answered Jul 21 at 17:12
Imari1234
562
It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^x^2$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.
A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.
answered Jul 21 at 17:12
Imari1234
562
answered Jul 21 at 17:12
Imari1234
562
answered Jul 21 at 17:12
Imari1234
562
answered Jul 21 at 17:12
Imari1234
562
562
add a comment |Â
add a comment |Â
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4
what makes you think that the function is integrable into elementary functions? Not every function has a "nice" integration.
– Doug M
Jul 19 at 23:23
3
There are a lot of expressions that don't have a nice formula for their antiderivative; e.g. $e^-x^2$. If this is simply a random expression, then this might be one of them. Do you have any other reason to believe it does have an antiderivative in terms of common functions?
– robjohn♦
Jul 19 at 23:23
@robjohn I didn't think about that... in that case, what other types of functions would be necessary?
– RayDansh
Jul 19 at 23:24
1
@RayDansh: the function $cosleft(x^3right)sin(x)$ is locally integrable, so an antiderivative exists. However, that anti-derivative may not be expressible in terms of any common functions. $int e^-x^2,mathrmdx$ can be expressed in terms of the erf function, whose definition is $operatornameerf(x)=frac2sqrtpiint_0^x e^-t^2,mathrmdt$. In this case, the function is simply defined as the integral we are seeking, so it is sort of circular.
– robjohn♦
Jul 19 at 23:58
1
If you need functions to antidifferentiate which are guaranteed to have an elementary solution, get a friend to choose a function, differentiate it and give you the result. Of course you have to trust your friend to do the differentiation correctly :)
– David
Jul 20 at 0:05