Solve the indefinite integral of 1/sqrt(x^2+2x+5).
I solved it all out by completing the square and then trig sub and then drawing a triangle to go back to x variable and got:
ln |(sqrt(x^2+2x+5)/2 + (X+1)/2 ) |
but the book answer is:
ln |(sqrt(x^2+2x+5) + x + 1 ) |
So, how did they rid of the 1/2 or where did I go wrong?
I don't seem to understand your explaination.
Factor x² + 2 x + 5:
x² + 2 x + 5 = ( x + 1 )² + 4
∫ dx / √ ( x² + 2 x + 5 ) = ∫ dx / √ [ ( x + 1 )² + 4 ]
Substitute:
( x + 1 ) / 2 = u
x + 1 = 2 u , dx = 2 du
∫ dx / √ [ ( x + 1 )² + 4 ] = ∫ 2 du / √ [ ( 2 u )² + 4 ] =
2 ∫ du / √ ( 4 u² + 4 ) =
2 ∫ du / √ [ 4 ∙ ( u² + 1 ) ] =
2 ∫ du / [ √4 ∙ √ ( u² + 1 ) ] =
2 ∫ du / [ 2 ∙ √ ( u² + 1 ) ] =
∫ du / √ ( u² + 1 ) = arc sinh ( u ) = arc sinh [ ( x + 1 ) / 2 ] + C
∫ dx / √ ( x² + 2 x + 5 ) = arcsinh [ ( x + 1 ) / 2 ] + C
So, the mysterious 2 crops again. However, for an indefinite integral, the extra constant does not matter, since
ln((x+...)/2) = -ln2 + ln(x+...) which disappears into the arbitrary constant C.
To find the indefinite integral of 1/sqrt(x^2+2x+5), you correctly started by completing the square and then making a trigonometric substitution. However, there seems to be a discrepancy in the simplification of the expression.
Let's go through the steps again:
1. Rewrite the integrand: 1/sqrt(x^2+2x+5).
2. Complete the square inside the square root:
x^2 + 2x + 5 = (x + 1)^2 + 4.
3. Make the substitution:
Let x + 1 = 2tan(theta), which implies dx = 2sec^2(theta) d(theta).
4. Substitute the trigonometric identity:
x = 2tan(theta) - 1.
Now, let's substitute these expressions back into the integral:
∫(1/sqrt(x^2+2x+5)) dx = ∫(1/sqrt((x+1)^2+4)) dx
= ∫(1/sqrt(4tan^2(theta)+4)) * 2sec^2(theta) d(theta)
= ∫(2sec^2(theta)/2sec(theta)) d(theta)
= ∫sec(theta) d(theta)
= ln |sec(theta) + tan(theta)| + C
= ln |(sqrt(x^2+2x+5) + x + 1 ) | + C.
After simplifying the result, we indeed have the book's answer: ln |(sqrt(x^2+2x+5) + x + 1 ) |.
It seems there might have been a calculation error or misinterpretation in simplifying the trigonometric substitution. Double-check your calculations and verify that the simplification mentioned above is correct.
This is how another person explained it to me:
dx/sqrt[(x+1)^2 +4 ]
let z = x+1
dx = dz
dz/sqrt[z^2 + 4]
= ln[z+sqrt(z^2+4)]
= ln[x+1 + sqrt(x^2+2x+1 + 4)]
because
integral dx/sqrt(x^2+p^2)
=ln[ x +sqrt(x^2+p^2)]
I still don't get how the 1/2 disappeared?
The question is, where did you get the 1/2 factor anyway? You have declined to show just what you did. You know that
x^2+2x+5 = (x+1)^2 + 2^2
and that
∫1/√(x^2+a^2) dx = ln(x+√(x^2+a^2))
since if
x = a*tanθ
x^2+a^2 = a^2 sec^2θ
dx = a sec^2θ dθ
∫1/√(x^2+a^2) dx = ∫1/(a secθ) (a sec^2θ) dθ
= ∫secθ dθ
= ln(secθ+tanθ) + C
So, where did you get your factor of 2?