In x²+1/x² how x²+1/x²+2 is coming
Answers
Answer:
EXPLANATION.
⇒ ∫1.dx/x² - 1.
As we know that,
First, we factorizes the denominator, we get.
x² - 1 it is in the form of (a² - b²) = (a + b)(a - b).
⇒ x² - 1 = (x - 1)(x + 1).
It is written as,
⇒ ∫dx/(x - 1)(x + 1) = A/(x - 1) + B/(x + 1).
⇒ 1 = A(x + 1) + B(x - 1).
Put the value of x = 1 in equation, we get.
⇒ 1 = A(1 + 1) + B(1 - 1).
⇒ 1 = 2A.
⇒ A = 1/2.
Put the value of x = -1 in equation, we get.
⇒ 1 = A(- 1 + 1) + B(- 1 - 1).
⇒ 1 = - 2B.
⇒ B = -1/2.
Put the value of A & B in equation, we get.
⇒ ∫A/(x - 1).dx + ∫B/(x + 1).dx.
⇒ ∫1/2/(x - 1).dx + ∫-1/2/(x + 1).dx.
⇒ ∫1.dx/2(x - 1) + ∫-1.dx/2(x + 1).
As we know that,
1/2 & -1/2 is a constant term it can out through integration, we get.
⇒ 1/2∫dx/(x - 1) - 1/2∫dx/(x + 1).
⇒ 1/2㏒(x - 1) - 1/2㏒(x + 1) + c.
MORE INFORMATION.
Important notes.
If a function can be expressed in terms of elementary function (formulae format) then only it is integral, other wise cannot.
For example :-
⇒ \sf \int e^{sin x} dx∫e
sinx
dx
⇒ ∫√sin(x)dx
⇒ ∫x⁴/x¹⁰ + 1
⇒ ∫Cos(x)/x dx
⇒ ∫dx/㏑(Sin(x)).