Integrating Powers of tanx | Trigonometry

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x^3+2x^2=3
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Honestly i would love to see some more videos abaut integrals on your main chanell or here because integrals are very intetesting topic

janekfortepianek

Good video. I just thought you would be working toward a general solution.

JefiKnight

I think it would be nice to generalize it like in the thumbnail. I generalized it for even and odd powers separately. I have been able to solve it for the odd powers directly, using the binominal theorem, and for the even powers, I found a recursive formula. I analyzed and estimated a explicit formula. I then prooved it via induction.

Dany-wi

2:30 Here I would use the fact that the first derivative of tan(x) = sin(x)/cos(x) can be written in two ways.
According to the quotient rule, the first derivative of tangens is
tan'(x)
= (cos(x)*cos(x) - sin(x)*(- sin(x)) / cos^2(x)
= (cos^2(x) + sin^2(x)) / cos^2(x)
= 1 / cos^2(x) (due to the addition theorem in the numerator) or
= 1 + tan^2(x), which is more useful in this case, since it directly connects the tangens to its derivative.
If we subtract 1, we get
tan^2(x) = tan'(x) - 1
and by integrating both sides:
integral tan^2(x) dx = tan(x) - x + const.
the same result as shown at 3:36.

goldfing

1:00 This kind of integral doesn't need a complicated substitution, because you see that the first derivative of cos(x) is -sin(x) and sin(x) is contained in the integrand, so you can just apply the chain rule for derivatives:
f(g(x))' = f'(g(x)) * g'(x)
and can set f(x) = ln(|x|), g(x) = cos(x), f'(x) = 1/x, g'(x) = -sin(x) and then write
f(g(x)) = ln(|cos(x)|)
f'(g(x))' * g'(x) = 1/cos(x) * (- sin(x)) = -sin(x)/cos(x) = -tan(x)
Therefore
integral tan(x) dx
= - integral -tan(x) dx
= -ln(|cos(x)|).

goldfing

very cool video

I have one question, what software do you use?

leonardobarrera

Great video on integration ! Just a question: which of your accounts prevail ?

paultoutounji

I enjoy your video please continue power of tanx

mahdiali

Dear friend, do you use tablet for write? Do you use some software ? It is excelent.🇧🇷🇧🇷🇧🇷🇧🇷

paulor.r.correia

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