How to convert sin2∅ in trigonometry substituion.

In summary, the question is asking for the integral of √(y^2-25)/y^3 dy. The given answer is 1/5 sec^-1(y/5)-sin2∅/20-∅/10, but the tutorial answer is 1/5 sec^-1(y/5)- √(y^2-25)/2y^2+c. To solve this, a trig substitution is used where sin^2 + cos^2 = 1. By letting z = y/SQRT(25) and dz = dy/5, the term involving the square root can be simplified to SQRT(z^2 - 1)/2z^2.
  • #1
clifftan
3
0
the question is ∫ √(y^2-25)/y^3 dy..

I did found the answer which is 1/5 sec^-1(y/5)-sin2∅/20-∅/10
But mine tutorial answer is 1/5 sec^-1(y/5)- √(y^2-25)/2y^2+c
i don't know how to convert the behind part sin2∅/20-∅/10 into √(y^2-25)/2y^2
 
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  • #2
The typical trig substitution is where sin^2 + cos^2 = 1.

Now you should let y^2 - 25 = (25/25)*y^2 - 25 = (25/25)*y^2 - 25 = 25[y^2/25 - 1]
= 25[(y^2/25 - 1].

Have a new variable where z = y/SQRT(25) (through an integral substitution) and you'll have to clean up a little but this term will go to [z^2 - 1] which can use a trig substitution of either sin or cos depending on what the other terms are like.

So to start off for your term involving the square root, factor out the 25 from the square root which will leave you with 5*SQRT(y^2/25 - 1)/2y^2 and make a substitution z = y/SQRT(25) which means dz = dy/5. Now clean up the square root sign to get 5*SQRT(z^2-1)/(2*25*z^2) * 5*dz = SQRT(z^2 - 1)/2z^2.
 
  • #3
Thank you very much~~
 

1. What is the formula for converting sin2∅ to trigonometry substitution?

The formula for converting sin2∅ to trigonometry substitution is sin2∅ = 2sin∅cos∅. This formula is derived from the double angle formula for sine, which states that sin2∅ = 2sin∅cos∅.

2. Why do we need to convert sin2∅ to trigonometry substitution?

Converting sin2∅ to trigonometry substitution allows us to simplify trigonometric expressions by using identities and formulas that are specific to double angles. This can make solving trigonometric equations and proving trigonometric identities much easier.

3. How do we use the substitution property to convert sin2∅ to trigonometry substitution?

The substitution property in trigonometry states that if two expressions are equal, then they can be substituted for each other in an equation or identity. To convert sin2∅ to trigonometry substitution, we use the substitution property to replace sin2∅ with 2sin∅cos∅.

4. Can we convert other trigonometric functions to trigonometry substitution?

Yes, other trigonometric functions can also be converted to trigonometry substitution using their respective double angle formulas. For example, cos2∅ can be converted to trigonometry substitution as 1-2sin²∅.

5. Are there any special cases when converting sin2∅ to trigonometry substitution?

Yes, there are two special cases to consider when converting sin2∅ to trigonometry substitution: when ∅ = 0 or ∅ = π/2. In these cases, the formula becomes 0 = 0 because sin0 = 0 and sin(π/2) = 1. This means that the original expression and the substitution are equivalent, and there is no need to continue simplifying.

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