Integrate using trig substitution

In summary: Remember to fill in the values of x and 4 when you are done.In summary, the problem is to find the integral of sqrt(8x-x^2) using the trigonometric substitution method. The solution involves completing the square and using the trigonometric substitution x-4=4sin(theta). After substituting and simplifying, the integral becomes int sqrt(4^2-(x-4)^2)dx. From here, the cosine and sine values can be found using the triangle method and substituted back in to solve the integral.
  • #1
n77ler
89
0

Homework Statement



int sqrt 8x-x^2

Homework Equations



trig sub

The Attempt at a Solution



complete the square
integral becomes
int sqrt 16-(x-4)^2

let x-4= 16sin(Q) sqrt 16-(x-4)^2 =sqrt 16-256sin^2(Q)
dx= 16cos(Q)dQ = 16cos(Q)


int 16cos(Q)16cos(Q) dQ=256 int cos^2(Q)dQ double angle formula

128 int 1+cos(2Q) dQ let u=2Q 1/2du=dQ

This is the part where i get messed up...
128 int 1 dQ + 128 int cos(u)du
So the first part is just 128Q and the integral of 128intcos(u)du is 128sin(u)
So do I fill 2Q back in for u? Then if I fill it back in do I use Q=[sqrt16-(x-4)^2] / 16 ??
 
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  • #2
[tex]\int\sqrt{8x-x^2}dx[/tex]

[tex]\int\sqrt{4^2-(x-4)^2}dx[/tex]

*[tex]x=a\sin\theta \rightarrow a=4[/tex]

[tex]x-4=4\sin\theta[/tex]
[tex]dx=4\cos\theta d\theta[/tex]

[tex]\int\sqrt{4^2-4^2\cos^2\theta}d\theta[/tex]

Re-check your substitutions!
 
  • #3
Ok so other than that does it look right? It is a definate integral so in the end i just plug my value in correct?
 
  • #4
I end up with 8[Q]+8[sin(2Q)] But I am not sure how to put a value back in for Q
 
  • #5
You want it in terms of x?

Use your substitutions ... cosine = adj/hyp and sine = opp/hyp.
 
  • #6
hmm I am not familiar with doing it like that
 
  • #7
Draw a triangle and you will know what I mean ...

[tex]x-4=4\sin\theta[/tex]

[tex]\sin\theta=\frac{x-4}{4}[/tex]

[tex](x-4)^2+\cos^2\theta=4^2[/tex]

[tex]\cos\theta=\frac{\sqrt{4^2-(x-4)^2}}{4}[/tex]
 
Last edited:

1. What is trig substitution and when is it used in integration?

Trig substitution is a method used to solve integrals that involve expressions with radicals or powers of trigonometric functions. It is typically used when the integrand contains expressions such as √(a^2 - x^2), √(x^2 + a^2), or √(x^2 - a^2).

2. How do I know when to use trig substitution?

You can use trig substitution when you have a radical expression or a power of a trigonometric function in the integrand. You can also use it when you have a quadratic expression in the integrand that can be simplified using trigonometric identities.

3. What are the steps to integrate using trig substitution?

The steps to integrate using trig substitution are:

  1. Identify which trigonometric substitution to use based on the form of the integrand.
  2. Make the appropriate substitution by replacing the trigonometric function with a variable.
  3. Simplify the resulting integral using trigonometric identities.
  4. Integrate the simplified expression using standard integration techniques.
  5. Substitute the original variable back into the final expression to get the solution.

4. Can trig substitution be used for all types of integrals?

No, trig substitution is only used for specific types of integrals that involve expressions with radicals or powers of trigonometric functions. It cannot be used for all types of integrals.

5. How can I check if my answer is correct when using trig substitution?

You can check your answer by differentiating it. If the derivative of your answer is equal to the original integrand, then your solution is correct. You can also use online tools or calculators to verify your answer.

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