What is #int ( 2 + x ) / sqrt ( 4 - 2x - x^2 dx#?

Question

What is #int  ( 2 + x ) / sqrt ( 4 - 2x - x^2   dx#?

Scarlett Allison · Accepted Answer

#int (2+x)/sqrt(4-2x-x^2)dx=arcsin ((x+1)/sqrt5)-sqrt(4-2x-x^2)+C#

Start from the given

#int (2+x)/sqrt(4-2x-x^2)dx#

Start with Algebra by completing the square

#4-2x-x^2=-(x^2+2x-4)=-(x^2+2x+1-1-4)#

and

#4-2x-x^2=-((x+1)^2-5)=5-(x+1)^2#

then

#int (2+x)/sqrt(4-2x-x^2)dx=int (2+x)/sqrt(5-(x+1)^2)dx#

The Trigonometric Substitution

Let #x+1=sqrt(5)*sin theta# and #x=sqrt(5)*sin theta -1# and #dx=sqrt(5)*cos d theta#

Let's do the substitution

#int (2+x)/sqrt(5-(x+1)^2)dx=# #int((sqrt(5)sin theta + 1)*(sqrt(5)*cos theta)d theta)/sqrt(5-(sqrt(5)*sin theta)^2)#

#int((sqrt(5)sin theta + 1)*(sqrt(5)*cos theta)d theta)/sqrt(5-5*sin^2 theta)#

continue simplification by trigonometric identities

#int((sqrt(5)sin theta + 1)*(sqrt(5)*cos theta)d theta)/(sqrt5*sqrt(1-sin^2 theta))#

#int((sqrt(5)sin theta + 1)*(sqrt(5)*cos theta)d theta)/(sqrt5*sqrt(cos^2 theta))#

#int((sqrt(5)sin theta + 1)*(sqrt(5)*cos theta)d theta)/(sqrt5*cos theta)#

and

#int (2+x)/sqrt(5-(x+1)^2)dx=int(sqrt(5)sin theta + 1)d theta#

#int (2+x)/sqrt(5-(x+1)^2)dx=int(1+sqrt(5)sin theta )d theta#

#int (2+x)/sqrt(5-(x+1)^2)dx=theta-sqrt(5)*cos theta+C#

Now, time to imagine your right triangle with angle #theta# Let #x+1# the Opposite side to angle #theta# Let #sqrt5# the Hypotenuse Let #sqrt(4-2x-x^2)# the Adjacent side to angle #theta#

Return the variables

#int (2+x)/sqrt(5-(x+1)^2)dx=theta-sqrt(5)*cos theta+C#

#int (2+x)/sqrt(5-(x+1)^2)dx=arcsin ((x+1)/(sqrt5))-sqrt(4-2x-x^2)+C#

I hope the explanation is useful....God bless...

Cameron Alexander · Answer

undefined The given expression is an integral. To solve it, you can follow these steps:

1. Rewrite the integral: ∫(2 + x) / sqrt(4 - 2x - x^2) dx.
2. Attempt to factorize the denominator to simplify the expression inside the square root.
3. Complete the square in the denominator if possible.
4. Use a trigonometric substitution or another suitable method to integrate the expression.

Without further context or constraints, a specific solution cannot be provided. Depending on the nature of the problem or additional instructions, different methods may be applicable to solve this integral.