Reverse Chain Rule
Can anyone help me integrate Sec^2x Tanx dx using reverse chain rule. Every time I do this I get 1/2 tan^2 x which is incorrect (the correct answer is 1/2 sec^2x. This is part of a larger question and the solution bank in the text book doesn’t go into depth on how to do this step. I see answers online using the substitution method but this question seems perfect for using reverse chain rule. Wondering if anyone can help?
Reply 1
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Original post
by ethandavisCan anyone help me integrate Sec^2x Tanx dx using reverse chain rule. Every time I do this I get 1/2 tan^2 x which is incorrect (the correct answer is 1/2 sec^2x. This is part of a larger question and the solution bank in the text book doesn’t go into depth on how to do this step. I see answers online using the substitution method but this question seems perfect for using reverse chain rule. Wondering if anyone can help?
Have you thought how those two expressions are related?
Reply 2
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Original post
by mqb2766Have you thought how those two expressions are related?
In this case, F(x) = tanx and F’(x) = sex^2x
That is why I thought this would be a perfect question using reverse chain rule since the rule is:
Integral of F’(x)(F(x))^n dx = (F(x))^n+1 /n+1
Reply 3
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Original post
by ethandavisIn this case, F(x) = tanx and F’(x) = sex^2x
That is why I thought this would be a perfect question using reverse chain rule since the rule is:
Integral of F’(x)(F(x))^n dx = (F(x))^n+1 /n+1
That is why I thought this would be a perfect question using reverse chain rule since the rule is:
Integral of F’(x)(F(x))^n dx = (F(x))^n+1 /n+1
No, how tan^2 and sec^2 are related, so the two answers.
Reply 4
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Original post
by mqb2766No, how tan^2 and sec^2 are related, so the two answers.
Tan^2x +1 = sex^2x but I can’t see how this would help me come to the correct answer that is 1/2 sex^2 x
Reply 5
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Original post
by ethandavisTan^2x +1 = sex^2x but I can’t see how this would help me come to the correct answer that is 1/2 sex^2 x
I agree so what is the difference between those two antiderivatives/integrals. So literally the difference.
Reply 6
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by mqb2766I agree so what is the difference between those two antiderivatives/integrals. So litererally the difference.
i’m not sure what you mean?
Reply 7
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by ethandavisi’m not sure what you mean?
If you subtract one from the other, literally the difference, what do you get?
Reply 8
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by mqb2766If you subtract one from the other, the difference, what do you get?
what functions are we subtracting?
Reply 9
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Original post
by ethandaviswhat functions are we subtracting?
The two answers/anti derivatives/integrals. So what is the difference between them. If you want to check whether one expression is similar/different to another expression, the usual is to subtract one from the other and see how theyre different.
Reply 10
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Original post
by mqb2766The two answers/anti derivatives/integrals. So what is the difference between them. If you want to check whether one expression is similar/different to another expression, the usual is to subtract one from the other and see how theyre different.
Im not sure what we are subtracting here. Ultimately Im trying to find the integral of sec^2 x tan x
To do this is set Y= tan^2 x
then found dy/dx which was 2tanxsec^2x
Therefore the integral of sec^2x tanx is = 1/2 tan^2x
Reply 11
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Original post
by ethandavisIm not sure what we are subtracting here. Ultimately Im trying to find the integral of sec^2 x tan x
To do this is set Y= tan^2 x
then found dy/dx which was 2tanxsec^2x
Therefore the integral of sec^2x tanx is = 1/2 tan^2x
To do this is set Y= tan^2 x
then found dy/dx which was 2tanxsec^2x
Therefore the integral of sec^2x tanx is = 1/2 tan^2x
What is
1/2 tan^2(x) - 1/2 sec^2(x)
the halves are not that important but Ive just put them in to make it clear the two terms are the two "answers".
Reply 12
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Original post
by mqb2766What is
1/2 tan^2(x) - 1/2 sec^2(x)
the halves are not that important but Ive just put them in to make it clear the two terms are the two "answers".
1/2 tan^2(x) - 1/2 sec^2(x)
the halves are not that important but Ive just put them in to make it clear the two terms are the two "answers".
Well since tan^2x - sec^x = 1
I would also say 1
Reply 13
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Original post
by ethandavisWell since tan^2x - sec^x = 1
I would also say 1
I would also say 1
Yes, so why is that not important (a constant) if the two terms are antiderivatives?
Reply 14
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by mqb2766Yes, so why is that not important (a constant) if the two terms are antiderivatives?
Im not too sure. I don’t understand why we are doing this subtraction in the first place. The correct answer to my original question was 1/2 sec^2x but I get 1/2 tan^2x from using the reverse chain rule.
Reply 15
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by ethandavisIm not too sure. I don’t understand why we are doing this subtraction in the first place. The correct answer to my original question was 1/2 sec^2x but I get 1/2 tan^2x from using the reverse chain rule.
First of all they are not the antiderivaties. Both antiderivatives should have a
+ c
Its important.
So there are a few ways to note 1/2tan^2(x)+c and 1/2sec^2(x)+c are both valid answers as you could start with
tan^2(x) + 1 = sec^2(x)
and note that when you differentiate both sides, the d(1)/dx=0 so the derivatives of both sides are the same.
Or
tan^2(x) + c = (1-cos^2(x))/cos^2(x) + c = sec^2(x) + C
or ....
So they represent identity transformations apart from a difference by a constant (hence the above check to difference them). Obv Ive been a bit lax with the 1/2s but its not that important.
Reply 16
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Ahhh, I see what you’re saying. This integration can have two solutions but due to the constant (c), these solutions are equal. Thank you very much!
Reply 17
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Original post
by ethandavisAhhh, I see what you’re saying. This integration can have two solutions but due to the constant (c), these solutions are equal. Thank you very much!
Sure, you could also have noted that the original integrand could be written as
(sec^2(x)) tan(x)
(sec(x)tan(x)) sec(x)
and doing the reverse chain rule on each leads directly to the two different, but equivalent, anti derivatives (remembering to put the +c in as well).
Its trig. You study doing identity transformations for a reason. But due to the pythagorean identities, sec and tan are closely related (calculus), just as sin and cos are and cosec and cot. You should be "question spotting" terms like that.
(edited 1 year ago)
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