Trigonometric Substitution problem

In summary, the conversation discusses the process of solving the integral ∫(√(x²-4))/x dx using the substitution x=2secθ, dx=2secθtanθdθ. The conversation also addresses a misunderstanding about the substitution for dx and provides the correct solution using the substitution. The final result is ∫(√(x²-4))/x dx = ln|x|-ln|2|-ln|x+√(x²-4)|+ln|2|+C.
  • #1
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1. [itex]∫\frac{\sqrt{x^{2}-4}}{x} dx[/itex], [itex]x=2secθ[/itex], [itex]dx=2secθtanθ dθ[/itex]



2. [itex]\sqrt{x^{2}-a^{2}}[/itex],[itex]sec^{2}θ-1=tan^{2}θ[/itex]



3. [itex]\sqrt{x^{2}-4}=\sqrt{4sec^{2}θ-4}=\sqrt{4(1+tan^{2}θ)-4}=\sqrt{4tan^{2}θ}=2\left|tanθ\right|=2tanθ[/itex];[itex]∫\frac{\sqrt{x^{2}-4}}{x}dx=∫\frac{2tanθ}{2secθ}dθ=\frac{ln\left|secθ\right|}{ln\left|secθ+tanθ\right|}+C=ln\left|secθ\right|-ln\left|secθ+tanθ\right|+C[/itex];[itex]∫\frac{\sqrt{x^{2}-4}}{x}dx=ln\left|\frac{x}{2}\right|-ln\left|\frac{x}{2}+\frac{\sqrt{x^{2}-4}}{2}\right|+C=ln\left|x\right|-ln\left|2\right|-ln\left|x+\sqrt{x^{2}-4}\right|+ln\left|2\right|+C[/itex]. Please tell me what I did wrong.
 
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  • #2
What happened to the substitution for dx in terms of dθ?
 
  • #3
I see what you mean. Ok, here is what I have after the dx substitution: [itex]\int\frac{\sqrt{x^{2}-4}}{x}dx = \int\frac{2tanθ}{2secθ}2secθtanθdθ = 2\int tan^{2}θdθ[/itex]
 
  • #4
Which is: [itex]2\int(sec^{2}θ-1)dθ = 2(tanθ-θ)+C[/itex]
 

1. What is Trigonometric Substitution?

Trigonometric substitution is a method used to solve integrals involving algebraic expressions and trigonometric functions. It involves substituting a trigonometric function for an algebraic expression, which can simplify the integral and make it more manageable to solve.

2. When should Trigonometric Substitution be used?

Trigonometric substitution should be used when solving integrals that involve expressions with square roots, or when the integrand contains a sum or difference of squares.

3. How do you choose the appropriate trigonometric substitution?

The substitution used depends on the form of the integral. For integrals with the form √(a2+x2), use the substitution x = a tan(θ). For integrals with the form √(x2-a2), use the substitution x = a sec(θ). For integrals with the form √(a2-x2), use the substitution x = a sin(θ).

4. What are the common mistakes made when using Trigonometric Substitution?

Some common mistakes include choosing the wrong substitution, not correctly rewriting the integral in terms of the substitution variable, and not substituting the limits of integration. It is also important to check for any trigonometric identities that may simplify the integral further.

5. Can Trigonometric Substitution be used for definite integrals?

Yes, Trigonometric Substitution can be used for definite integrals. However, it is important to substitute the limits of integration in terms of the substitution variable and evaluate the integral using these new limits.

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