How Can I Integrate x*sqrt(1-x^4) Using Trig Substitution?

In summary, a trigonometric substitution is a technique used in integration to transform an integral involving algebraic expressions into one involving trigonometric functions. It is commonly used when dealing with square roots of quadratic expressions or when the integrand contains certain trigonometric functions. The three main types of trigonometric substitutions are sine, cosine, and tangent substitutions. To make a successful trigonometric substitution, one must identify the appropriate type of substitution, choose an appropriate trigonometric function, and simplify the integral using trigonometric identities. When solving integrals with trigonometric substitutions, it is important to check the limits of integration and practice identifying the appropriate type of substitution.
  • #1
annie122
51
0
how do i go about integrating

x*sqrt(1-x^4)??

i have no idea
 
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  • #2
Re: another trig substitution

Yuuki said:
how do i go about integrating

\(\displaystyle \int x \cdot \sqrt{1-x^4}dx\)

i have no idea

I would let \(\displaystyle u = x^2 \therefore du = 2x dx\)

\(\displaystyle \int x \cdot \sqrt{1-x^4} \cdot \dfrac{du}{2x} = \dfrac{1}{2} \int \sqrt{1-u^2} du\)

Do you know a trig sub for this integral?
 
  • #3
Re: another trig substitution

u = sin(w) right?

thx!
 
  • #4
Re: another trig substitution

Yuuki said:
u = sin(w) right?

thx!

Yes, that's the right sub to use. Are you ok going forward?
 
  • #5
yup thank you
 

1. What is a trigonometric substitution?

A trigonometric substitution is a technique used in integration to transform an integral involving algebraic expressions into one involving trigonometric functions. This is useful for solving integrals that cannot be solved by other methods.

2. How do I know when to use a trigonometric substitution?

Trigonometric substitutions are most commonly used when dealing with square roots of quadratic expressions, or when the integrand contains certain trigonometric functions such as sine, cosine, or tangent.

3. What are the three main types of trigonometric substitutions?

The three main types of trigonometric substitutions are:

  1. Sine substitution: used when the integrand contains √(a² - x²)
  2. Cosine substitution: used when the integrand contains √(a² + x²)
  3. Tangent substitution: used when the integrand contains √(x² - a²)

4. How do I make a successful trigonometric substitution?

When making a trigonometric substitution, you need to identify which type of substitution to use based on the form of the integrand. Then, you need to choose an appropriate trigonometric function to replace the variable in the integrand. Finally, you need to manipulate the integral using trigonometric identities to simplify it into a form that can be easily integrated.

5. Are there any tips for solving integrals with trigonometric substitutions?

One tip for solving integrals with trigonometric substitutions is to always check the limits of integration. This may require you to use a trigonometric identity to rewrite the limits in terms of the new variable. Additionally, it is helpful to practice identifying which type of trigonometric substitution to use for different types of integrals.

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