Trig Integration By Substitution

In summary, the given conversation discusses the integration of (2x+6)/sqrt(5-4x-x^2) using trigonometric substitutions. The final substitution used is u= 3 sin(theta) and the resulting integral is 2/3(ln|tan(theta)+sec(theta)|-3|cos(theta)|).
  • #1
Rachael95
2
0
Mod note: Moved from technical math section
∫(2x+6)/sqrt(5-4x-x^2)

I have 2/3(ln|tan(theta)+sec(theta)|-3|cos(theta)|) where x=sin^-1((x+2)/3)
 
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  • #2
Rachael95 said:
∫(2x+6)/sqrt(5-4x^2-x^2)

Are there supposed to be two x^2 expressions under the SQRT?
 
  • #3
Apologies no it should be 5-4x-x^2
 
  • #4
Complete the square in the square root: [tex]5- 4x- x^2= 5- (x^2+ 4x+ 4- 4)= 5- (x+ 2)^2+ 4= 9- (x+ 2)^2.

Now make the substitution u= x+ 2, du= dx, x= u- 2 so 2x+ 6= 2u+ 2. The integral becomes [tex]\int \frac{2u+ 2}{\sqrt{9- u^2} du[/tex]

Now let [itex]u= 3 sin(\theta)[/itex].
 
  • #5
Homework-type problems should be posted in the Homework & Coursework section. I have moved this thread.
 

1. What is trig integration by substitution?

Trig integration by substitution is a method used to solve integrals involving trigonometric functions. It involves substituting a variable with a trigonometric expression to simplify the integral.

2. How do you know when to use trig integration by substitution?

You can use trig integration by substitution when the integral involves a trigonometric function and its derivative, and when the integral cannot be solved using traditional integration techniques.

3. What are the steps for performing trig integration by substitution?

The steps for trig integration by substitution are: 1) Identify the trigonometric function and its derivative in the integral, 2) Choose a substitution variable that will eliminate the trigonometric function, 3) Substitute the variable and its derivative into the integral, 4) Solve the new integral, 5) Substitute back the original variable to get the final solution.

4. Can you provide an example of trig integration by substitution?

Sure! Let's say we have the integral ∫ cos(2x) dx. We can use the substitution u = 2x, du = 2dx to rewrite the integral as ∫ cos(u) (1/2)du. Solving this integral gives us (1/2)sin(u) + C. Substituting back u = 2x, we get the final solution (1/2)sin(2x) + C.

5. Are there any tips for solving more complex trig integration by substitution problems?

Yes, here are a few tips: 1) Choose a substitution that will eliminate the most complicated trigonometric function, 2) Consider using trigonometric identities to simplify the integral before substituting, 3) Check your answer by differentiating it to make sure it matches the original integrand.

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