Integrate sqrt(1-x^4)/x^5 dx Using Trig Sub | Yahoo Answers

In summary, to integrate sqrt(1-x^4)/x^5 dx using trig sub, we first make the substitution x^2 = cos(theta) and then use integration by parts to obtain the final result of I = (1/4)*ln((sqrt(1-x^4)+1)/x^2) - sqrt(1-x^4)/x^4 + C.
  • #1
MarkFL
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Here is the question:

How do I integrate sqrt(1-x^4)/x^5 dx using trig sub?

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello The_Reporter,

We are given to compute the indefinite integral:

\(\displaystyle I=\int\frac{\sqrt{1-x^4}}{x^5}\,dx\)

Let's try the substitution:

\(\displaystyle x^2=\cos(\theta)\,\therefore\,dx=-\frac{\sin(\theta)}{2x}\,d\theta\)

And we obtain:

\(\displaystyle I=-\frac{1}{2}\int\frac{\sin^2(\theta)}{\cos^3(\theta)}\,d\theta\)

Next, let's try integration by parts where:

\(\displaystyle u=\sin(\theta)\,du=\cos(\theta)\,d\theta\)

\(\displaystyle dv=\frac{\sin(\theta)}{\cos(\theta}\,d\theta\,\therefore\,v=\frac{1}{2\cos^2(\theta)}\)

And now we obtain:

\(\displaystyle I=-\frac{1}{4}\left(\frac{\sin(\theta)}{\cos^2(\theta)}-\int\sec(\theta)\,d\theta\right)\)

For the remaining integral, consider:

\(\displaystyle \sec(\theta)=\frac{\sec(\theta)\left(\sec(\theta)+\tan(\theta)\right)}{\sec(\theta)+\tan(\theta)}=\frac{d}{d\theta}\left(\ln\left|\sec(\theta)+\tan(\theta)\right|\right)\)

Hence, we obtain:

\(\displaystyle I=-\frac{1}{4}\left(\frac{\sin(\theta)}{\cos^2(\theta)}-\ln\left|\sec(\theta)+\tan(\theta)\right|\right)+C\)

From our original substitution, we find that:

\(\displaystyle \tan(\theta)=\frac{\sqrt{1-x^4}}{x^2}\)

\(\displaystyle \sec(\theta)=\frac{1}{x^2}\)

And so, we finally find:

\(\displaystyle \bbox[5px,border:2px solid #207498]{I=\frac{1}{4}\left(\ln\left(\frac{\sqrt{1-x^4}+1}{x^2} \right)-\frac{\sqrt{1-x^4}}{x^4} \right)+C}\)
 

FAQ: Integrate sqrt(1-x^4)/x^5 dx Using Trig Sub | Yahoo Answers

What is the formula for integrating sqrt(1-x^4)/x^5 dx using trig substitution?

The formula for integrating sqrt(1-x^4)/x^5 dx using trig substitution is: ∫ sqrt(1-x^4)/x^5 dx = -1/8∫ sec^3θ dθ

How do you choose the appropriate trig substitution for this integral?

To choose the appropriate trig substitution for this integral, you need to identify the form of the integral. In this case, the form is ∫ sqrt(1-x^4)/x^5 dx, which indicates that the integral can be solved using a trigonometric substitution with the substitution x = tanθ.

What are the limits of integration for this integral?

The limits of integration for this integral depend on the substitution chosen. In this case, since x = tanθ, the limits of integration will be from 0 to π/4.

What is the purpose of using trig substitution in this integral?

The purpose of using trig substitution in this integral is to simplify the expression and make it easier to integrate. By substituting x with a trigonometric function, the integral can be transformed into a simpler form that can be easily solved using integration techniques.

Are there any special cases or restrictions when using trig substitution for this integral?

Yes, there are a few special cases and restrictions to keep in mind when using trig substitution for this integral. These include:

  • The integral must have a radical expression in the numerator.
  • The integral must contain a power of x in the denominator.
  • The integral must be a proper rational function.
  • The limits of integration must be in terms of the substitution variable.

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