Integral $int frac{cos(x)+sin(2x)}{sin(x)}text{ d}x$
up vote
3
down vote
favorite
I would like to evaluate
$$int dfrac{cos(x)+sin(2x)}{sin(x)}text{ d}xtext{.}$$
This is from Stewart's Calculus text, section 7.2., #19. Please note that I don't have a solutions manual and I am not interested in a complete solution.
I would merely like some guidance on a first step on approaching this one, since I've gotten 9 similar problems correct, and I hit a brick wall on this one. My first thought was perhaps splitting the fraction like so:
$$intcot(x)text{ d}x + intdfrac{sin(2x)}{sin(x)}text{ d}x$$
but this does not look helpful.
calculus integration indefinite-integrals trigonometric-integrals
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
add a comment |
up vote
3
down vote
favorite
I would like to evaluate
$$int dfrac{cos(x)+sin(2x)}{sin(x)}text{ d}xtext{.}$$
This is from Stewart's Calculus text, section 7.2., #19. Please note that I don't have a solutions manual and I am not interested in a complete solution.
I would merely like some guidance on a first step on approaching this one, since I've gotten 9 similar problems correct, and I hit a brick wall on this one. My first thought was perhaps splitting the fraction like so:
$$intcot(x)text{ d}x + intdfrac{sin(2x)}{sin(x)}text{ d}x$$
but this does not look helpful.
calculus integration indefinite-integrals trigonometric-integrals
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
4
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I would like to evaluate
$$int dfrac{cos(x)+sin(2x)}{sin(x)}text{ d}xtext{.}$$
This is from Stewart's Calculus text, section 7.2., #19. Please note that I don't have a solutions manual and I am not interested in a complete solution.
I would merely like some guidance on a first step on approaching this one, since I've gotten 9 similar problems correct, and I hit a brick wall on this one. My first thought was perhaps splitting the fraction like so:
$$intcot(x)text{ d}x + intdfrac{sin(2x)}{sin(x)}text{ d}x$$
but this does not look helpful.
calculus integration indefinite-integrals trigonometric-integrals
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
I would like to evaluate
$$int dfrac{cos(x)+sin(2x)}{sin(x)}text{ d}xtext{.}$$
This is from Stewart's Calculus text, section 7.2., #19. Please note that I don't have a solutions manual and I am not interested in a complete solution.
I would merely like some guidance on a first step on approaching this one, since I've gotten 9 similar problems correct, and I hit a brick wall on this one. My first thought was perhaps splitting the fraction like so:
$$intcot(x)text{ d}x + intdfrac{sin(2x)}{sin(x)}text{ d}x$$
but this does not look helpful.
calculus integration indefinite-integrals trigonometric-integrals
calculus integration indefinite-integrals trigonometric-integrals
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
edited Nov 27 at 12:15
Martin Sleziak
44.6k7115269
44.6k7115269
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
asked Oct 17 '14 at 4:02
Clarinetist
10.8k42777
10.8k42777
4
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05
add a comment |
4
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05
4
4
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05
add a comment |
1 Answer
1
active
oldest
votes
up vote
7
down vote
accepted
Substituting using the double-angle identity
$$sin (2x) = 2 sin x cos x$$
will transform the integrand into an expression that involves only $sin x$ and $cos x$, which suggests a particular substitution.
answered Oct 17 '14 at 4:04
Travis
59.1k766144
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
|
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
Substituting using the double-angle identity
$$sin (2x) = 2 sin x cos x$$
will transform the integrand into an expression that involves only $sin x$ and $cos x$, which suggests a particular substitution.
answered Oct 17 '14 at 4:04
Travis
59.1k766144
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
|
show 1 more comment
up vote
7
down vote
accepted
Substituting using the double-angle identity
$$sin (2x) = 2 sin x cos x$$
will transform the integrand into an expression that involves only $sin x$ and $cos x$, which suggests a particular substitution.
answered Oct 17 '14 at 4:04
Travis
59.1k766144
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
|
show 1 more comment
up vote
7
down vote
accepted
up vote
7
down vote
accepted
Substituting using the double-angle identity
$$sin (2x) = 2 sin x cos x$$
will transform the integrand into an expression that involves only $sin x$ and $cos x$, which suggests a particular substitution.
answered Oct 17 '14 at 4:04
Travis
59.1k766144
Substituting using the double-angle identity
$$sin (2x) = 2 sin x cos x$$
will transform the integrand into an expression that involves only $sin x$ and $cos x$, which suggests a particular substitution.
answered Oct 17 '14 at 4:04
Travis
59.1k766144
answered Oct 17 '14 at 4:04
Travis
59.1k766144
answered Oct 17 '14 at 4:04
Travis
59.1k766144
answered Oct 17 '14 at 4:04
Travis
59.1k766144
59.1k766144
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
|
show 1 more comment
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
1
1
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
You don’t even need a substitution: after using the double-angle formula, separate the integrand into two fractions, each easily integrated.
– Lubin
Oct 17 '14 at 4:12
1
1
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin Well, the integral of $cot x$ is not that easy to remember. I usually recognise the integrand as of the form $frac{(sin x)'}{sin x}$, but that's basically equivalent to substituting $u = sin x$.
– Deepak
Oct 17 '14 at 4:14
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Lubin You're quite right, of course, but when a student first encounters this type of integral they may well have only seen integrands involving $sin$ and $cos$ and in particular may not yet have seen how to handle $int cot x ,dx$.
– Travis
Oct 17 '14 at 4:18
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
@Travis - Thank you for your suggestion. I'm a recent math graduate who's studying for the Math GRE Subject Test, and I want to keep memorization to a minimum, so I appreciate yor answer.
– Clarinetist
Oct 17 '14 at 4:19
2
2
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
@Lubin I think it is well-known to everyone who has learned calculus, but it may not be obvious to people who are only just learning about trig integrals.
– Travis
Oct 17 '14 at 4:36
|
show 1 more comment
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4
hint : $sin (2x) = 2sin( x) cos (x)$
– ganeshie8
Oct 17 '14 at 4:03
A tip for solving integrals that have sin or cosine or some trigonometric function is use a bunch of identities
– Joao
Oct 17 '14 at 4:05