It might simple to you, but i'm stuck.

[lpheng]

The question is something like this, find the:
d/dx of sin^-1 (X+a) /b.
then evaluate,
intergarte of X / (9-4x-x²) ^1/2
&
intergarte of X² / (9-4x-x²) ^1/2


What i get for the first part is,
d/dx of sin^-1 (X+a) /b = 1 / ( -X²/b²-2aX/b²-a²+b²/b²)^1/2


Then I'm stuck. xD
Sorry for the confusing equation.

 

[tiny-tim]

welcome to pf!

hi lpheng! welcome to pf!

fiddle about with the a and b in your first answer until you get 9-4x-x²

 

[matphysik]

The question is something like this, find the:
d/dx of sin^-1 (X+a) /b.
then evaluate,
intergarte of X / (9-4x-x²) ^1/2
&
intergarte of X² / (9-4x-x²) ^1/2What i get for the first part is,
d/dx of sin^-1 (X+a) /b = 1 / ( -X²/b²-2aX/b²-a²+b²/b²)^1/2Then I'm stuck. xD
Sorry for the confusing equation.

With y=sin⁻¹[(x+a)/b], dy/dx= 1/√[b² - x² - 2xa - a²]

Let ℐ≡ ∫ xdx/√(9-4x-x²). Set u:= 4x+x²⇒½du-2dx=xdx. Whence ℐ= ∫ ½du/√(9-u) - 2sin⁻¹[(x+2)/√5] + constant. And so on...

NOTE1: 2a=4 ⇒ a=2 and b² - a²=9 ⇒ b=√5.

NOTE2: Given, ½du-2dx=xdx multiply through by `x` to get ½xdu-2xdx=x²dx.

 

[matphysik]

Let ℐ≡ ∫ xdx/√(9-4x-x²). Set u:= 4x+x²⇒½du-2dx=xdx. Whence ℐ= ∫ ½du/√(9-u) - 2sin⁻¹[(x+2)/√5] + constant. And so on...

NOTE1: 2a=4 ⇒ a=2 and b² - a²=9 ⇒ b=√5.

Correction: b² - a²=9 ⇒ b=√13. So that,

ℐ= ∫ ½du/√(9-u) - 2sin⁻¹[(x+2)/√13] + constant.

 

[blue_raver22]

No need to apologize for the equation, it is a common occurrence in scientific work to come across complex equations that can be confusing. Let me try to break down the steps for you to find the derivative and integrate the given equations.

For the first part, you have correctly found the derivative of sin-1 (X+a) /b. To evaluate it, you can simplify the expression by expanding the denominator and then simplifying it further. Once you have simplified it, you can then substitute the value of x in the expression and evaluate it to get the final result.

For the second and third parts, you can use the substitution method to integrate the given equations. You can substitute u = 9-4x-x2 and then use the formula for integration to solve for the final result. Make sure to substitute back the value of u at the end to get the final answer in terms of x.

I hope this helps you in solving the equations. Remember, when stuck, it is always helpful to break down the steps and try to simplify the equation before proceeding. Good luck!