INTEGRATION OF RATIONAL FUNCTION WITH SQUARE ROOT IN DENOMINATOR

Question 1 :

Evaluate the following with respect to "x".

(x + 2) / √(x² - 1)

Solution :

(x + 2) / √(x² - 1) = ∫(x/√(x² - 1)) dx + 2∫(1/√(x² - 1)) dx

∫(x/√(x² - 1)) dx

x² - 1 = t

2x dx = dt

x dx = dt/2

∫(x/√(x² - 1)) dx = ∫(1/√t) dt

= ∫t^-1/2 (dt/2)

= (1/2) t^1/2/(1/2)

= √(x² - 1) ---(1)

2∫(1/√(x² - 1)) dx = 2 log(x + √(x² - 1)) ------(2)

(1) + (2)

= √(x² - 1) + 2 log(x + √(x² - 1)) + c

Question 2 :

Evaluate the following with respect to "x".

(2x + 3) / √(x²+ 4x + 1)

Solution :

∫(2x + 3) / √(x²+ 4x + 1) dx

(2x + 3) = A(d/dx) (x²+ 4x + 1) + B

2x + 3 = A (2x + 4) + B ----(1)

Equating the coefficients of x.

2 = 2A

A = 1

Equating constant terms

3 = 4A + B

3 = 4(1) + B

3 = 4 + B

B = 3 - 4 ===> B = -1

Applying the values of A and B in (1)

2x + 3 = 1 (2x + 4) - 1

By dividing each term by √(x²+ 4x + 1), we get

∫(2x + 1) / √(x²+ 4x + 1) dx

= ∫(2x+4)/√(x²+ 4x + 1) dx - 1 ∫1/√(x²+ 4x + 1)dx

= 2 log √(x²+ 4x + 1) - 1 ∫1/√(x²+ 4x + 1)dx

√(x²+ 4x + 1) = [x²+ 2x(2) + 2² - 2² + 1]

= [(x + 2)²- 3]

= [(x + 2)²- √3²]

= 2 log √(x²+ 4x + 1) - 1 ∫1/√[(x + 2)²- √3²] dx

= 2 log √(x²+ 4x + 1) - log (x + 2) + √(x²+ 4x + 1) + c

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