Integral- Without using integration technnique

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    Integral Integration
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Discussion Overview

The discussion revolves around the integral \(\int \tan(t) \sec^3(t) dt\) and explores methods to solve it without employing traditional integration techniques such as substitution or integration by parts. Participants express varying opinions on the feasibility of this approach.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the meaning of "without using integration techniques," suggesting that integration techniques are inherently necessary to solve an integral.
  • Another participant states that the integral in question is not elementary, implying that it cannot be solved without some form of integration technique.
  • A different participant proposes a potential solution by suggesting that if \(f(x) = \frac{1}{3} \sec^3(\theta) + C\), then the derivative \(f'(x)\) leads back to the original integral, although this approach still relies on integration concepts.
  • Another participant notes that the power rule cannot be applied directly without substitution, indicating a limitation in the proposed methods.

Areas of Agreement / Disagreement

Participants express disagreement regarding the possibility of solving the integral without integration techniques. Some believe it is not feasible, while others attempt to propose solutions that still reference integration concepts.

Contextual Notes

The discussion highlights limitations in the proposed methods, particularly the reliance on integration concepts despite attempts to avoid them. There is also an acknowledgment of the integral's non-elementary nature.

Chipset3600
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Hello MHB, how can i solve this without use integration technniques...

\int tan(t)sec^3(t)dt
 
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Re: Integral- Without use integration technnique

Chipset3600 said:
Hello MHB, how can i solve this without use integration technniques...

\int tan(t)sec^3(t)dt

How do you mean "without using integration techniques"? Surely you need integration techniques to solve an integral? Also what have you tried?

Hint: Let $u = \sec(t) = \frac{1}{\cos(t)}$
 
Re: Integral- Without use integration technnique

SuperSonic4 said:
How do you mean "without using integration techniques"? Surely you need integration techniques to solve an integral? Also what have you tried?

Hint: Let $u = \sec(t) = \frac{1}{\cos(t)}$

I mean without: substitution, integration by parts...
 
Re: Integral- Without use integration technnique

I'm not sure that is possible, this is not an elementary integral (in the sense that it is the derivative of an elementary function, such as $\cos x$, $\sin x$, $\tan x$ and so on).
 
Re: Integral- Without use integration technnique

Hello, Chipset3600!

How can i solve this without use integration technniques?

. . \int \tan\theta \sec^3\!\theta\,d\theta
Well, maybe you can see all this?If we have: .f(x) \:=\:\tfrac{1}{3}\sec^3\!\theta + C

Then: .f'(x) \:=\:\tfrac{1}{3}\cdot 3\sec^2\!\theta\cdot\sec\theta\tan\theta + 0 \;=\;\tan\theta\sec^3\!\theta
 
Re: Integral- Without use integration technnique

soroban said:
Hello, Chipset3600!


Well, maybe you can see all this?If we have: .f(x) \:=\:\tfrac{1}{3}\sec^3\!\theta + C

Then: .f'(x) \:=\:\tfrac{1}{3}\cdot 3\sec^2\!\theta\cdot\sec\theta\tan\theta + 0 \;=\;\tan\theta\sec^3\!\theta

\int tan(t).sec^3(t)dt = \int sec^2(t).sec(t).tan(t)dtUsing the power rule now:\frac{sec^3(t)}{3}+C
 
Re: Integral- Without use integration technnique

You don't have a polynomial to apply the power rule, at least not until you use substitution.
 

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