Another couple of integration porblems

In summary, the conversation revolves around two integration problems - sech(1/x)tanh(1/x) all over x^2 and cos^-3(2θ)sin(2θ)dθ. The individual is having trouble with integrating hyperbolics and using properties for cubed and double angles. They have found a potential solution for the first problem through online research and are wondering if it is the correct approach. They also inquire about a possible substitution for the second problem. After receiving clarification and suggestions from another person, they express their gratitude.
  • #1
Back for a couple more I'm having a little trouble with. The first is

sech(1/x)tanh(1/x) all over x^2

Here I'm not sure how to deal with integrating hyperbolics at all as we haven't gotten to it in class yet. I did some searching on the internet and found that the integral tanh(x)sech(x) = -sech(x) + C. With all of it being over x^2 it becomes a little more complicated.

Do I rewrite x^2 as x^-2 and multiply it through with the 1/x's? Would the property tanh(x)sech(x) = -sech(x) + C still work?

I was also wondering about rewriting the whole equation as (2/e^1/x + e^-1/x)(e^1/x - e^-1/x /e^1/x - e^-1/x)(1/x^2). That seems overly complicated though and I'm not sure what form to go to from that point.

The other problem I'm having trouble with is integrating

cos^-3(2θ)sin(2θ)dθ

I know of ways of integrating sin and cos that have squares but is there a property I can use that's cubed and a double angel?
 
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  • #2
What do you mean by integrating all over x^2? I suppose you mean [tex]\int \text{sech}\left(\frac{1}{x}\right)\tanh\left(\frac{1}{x}\right)\frac{1}{x^2}dx[/tex]. If so use the substitution [itex]u=sech \left(\frac{1}{x}\right)[/itex]. For your second problem use the substitution [itex]u=\cos 2\theta[/itex]
 
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  • #3
Yeah, that is what I meant and both make so much more sense to me now. Thanks!
 

1. What is integration and why is it important in science?

Integration is the process of finding the area under a curve, or the accumulation of a quantity over a given interval. It is important in science because it allows us to calculate important physical quantities such as velocity, acceleration, and volume.

2. What is the difference between definite and indefinite integration?

Definite integration involves finding the exact numerical value of the area under a curve, while indefinite integration involves finding the general antiderivative of a function. Definite integration requires limits of integration, while indefinite integration does not.

3. How do I know which integration technique to use?

The choice of integration technique depends on the form of the integrand. Some common techniques include u-substitution, integration by parts, and trigonometric substitution. Practice and familiarity with these techniques will help you determine which one to use.

4. What are some common mistakes to avoid when integrating?

Some common mistakes include forgetting to include the constant of integration, mixing up the limits of integration, and incorrectly applying integration techniques. It is important to double check your work and practice regularly to avoid these mistakes.

5. Can integration be used in other fields besides mathematics?

Yes, integration is a fundamental concept that is used in various fields such as physics, engineering, economics, and biology. It allows us to model real-world phenomena and make predictions based on mathematical calculations.

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