Simple? Integration Problem (Trig Sub?)

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In summary, the problem asks for an equation in which the derivative of sin(t) with respect to t is multiplied by the derivative of cos(t) with respect to t. After differentiating both sides of the equation, the result is that the integral of sin(t)^2 with respect to t is equal to the integral of cos(t)^2 with respect to t.
  • #1
PitchBlack
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I can't get it! I'm pretty sure it's trig substition

[tex]\int[/tex][tex]x^{2}/\sqrt{1-x^{2}}[/tex]

Its a practice problem, if someone could show me the light (or steps) that would be wonderful
 
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  • #2
well, yeah a trig substitution would work, try to let x=sin(t) , so you will get dx=cos(t)dt
after you substitute it back you willl end up with something like this

integ of (sin(t))^2dt

Is this a homework problem by the way?

Can you go from here, anyway??
 
  • #3
No its not a homework problem...its a conceptual problem ...but i don't get it, and I'm not that great with calculus to tell you the truth I am not a math major i just want to get it! So is this what you mean...

[tex]\int(sinx)^{2}[/tex][tex] / [/tex][tex]\sqrt{1-sinx^{2}}[/tex]

so using u subtitution ( or whatever letter you use)...
u=sinx
du= cosx
and since there is no cos in the original then 1/cos(du)

(1/cos)[tex]\int du(u)^{2}[/tex][tex] / [/tex][tex]\sqrt{1-u^{2}}[/tex]

and go from there? did i do it right?
 
  • #4
PitchBlack said:
No its not a homework problem...its a conceptual problem ...but i don't get it, and I'm not that great with calculus to tell you the truth I am not a math major i just want to get it! So is this what you mean...

[tex]\int(sinx)^{2}[/tex][tex] / [/tex][tex]\sqrt{1-sinx^{2}}[/tex]

so using u subtitution ( or whatever letter you use)...
u=sinx
du= cosx
and since there is no cos in the original then 1/cos(du)

(1/cos)[tex]\int du(u)^{2}[/tex][tex] / [/tex][tex]\sqrt{1-u^{2}}[/tex]

and go from there? did i do it right?
Well You did not get it right, to be honest. Look, [tex]\int\frac{x^{2}}{\sqrt{1-x^{2}}}dx[/tex] now let sin(t)=x, from here after defferentiating we get cos(t)dt=dx, now let us substitute this back to the integral, so the integral will take this form:
[tex]\int\frac{(sin(t))^{2}}{\sqrt{1-(sin(t))^{2}}}cos(t)dt[/tex], now remember that (sin(t))^2= 1-(cos(t))^2, so afer we substitute the integral becomes:
[tex]\int\frac{(sin(t))^{2}}{\sqrt{(cos(t))^{2}}}cos(t)dt[/tex]= [tex]\int\frac{(sin(t))^{2}}{cos(t)}cos(t)dt[/tex]= [tex]\int (sin(t))^{2}dt[/tex], now do u know how to evaluate this one?
 
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Related to Simple? Integration Problem (Trig Sub?)

1. What is a "Simple? Integration Problem"?

A "Simple? Integration Problem" refers to a type of mathematical problem in which a function needs to be integrated using basic techniques. These techniques include substitution, integration by parts, and trigonometric substitution. These problems are typically encountered in calculus courses and involve finding the area under a curve or the volume of a solid.

2. What is a Trig Substitution?

A Trig Substitution is a method used to solve integration problems that involve trigonometric functions. It involves substituting a complex expression in the integrand with a simpler trigonometric expression, making it easier to integrate. Trig substitution is often used when the integral involves expressions such as √(a^2 - x^2), √(x^2 - a^2), or √(x^2 + a^2).

3. How do you solve a "Simple? Integration Problem" using Trig Substitution?

To solve a "Simple? Integration Problem" using Trig Substitution, follow these steps:
1. Identify the type of trigonometric function present in the integrand.
2. Substitute the trigonometric function with a new variable using a trig identity.
3. Use algebra to simplify the integrand.
4. Integrate the simplified expression.
5. Substitute the original variable back into the solution.
6. Simplify and solve for the constant of integration, if necessary.

4. Can all integration problems be solved using Trig Substitution?

No, not all integration problems can be solved using Trig Substitution. Trig substitution is most effective when the integrand contains a radical expression involving trigonometric functions. Other integration techniques, such as substitution and integration by parts, may be more suitable for different types of integrals.

5. What are some common mistakes made when using Trig Substitution to solve integration problems?

Some common mistakes when using Trig Substitution include:
- Choosing the wrong trigonometric substitution
- Forgetting to substitute back the original variable
- Making algebraic errors when simplifying the integrand
- Forgetting to include the constant of integration
- Not double-checking the final solution with the original integrand

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