Question

Calculeaza integrala nedefinita: ∫√x²-2x-1 dx

Answer 1

∫{(x² - 2x - 1) /[(x - 1)²(x²+ 1)]} dx ⇒

⇒(x² - 2x - 1) /[(x - 1)²(x²+ 1)] = A/(x - 1) + B/(x - 1)² + (Cx + D)/(x²+ 1) ⇒

⇒(x² - 2x - 1) /[(x - 1)²(x²+ 1)] ⇒

⇒[A(x - 1)(x²+ 1) + B(x²+ 1) + (Cx + D)(x - 1)²] /[(x - 1)²(x²+ 1)] ⇒

⇒(x² - 2x - 1) = [A(x - 1)(x²+ 1) + B(x²+ 1) + (Cx + D)(x - 1)²] ⇒

⇒x² - 2x - 1 = A(x³+ x - x²- 1) + Bx²+ B + (Cx + D)(x² - 2x + 1) ⇒

⇒x² - 2x - 1 = Ax³+ Ax - Ax²- A + Bx²+ B + Cx³ - 2Cx² + Cx + Dx² - 2Dx + D⇒

⇒x² - 2x - 1 = (A + C)x³+ (- A + B - 2C + D)x² + (A + C - 2D)x + (- A + B + D)⇒

{ A + C = 0
{- A + B - 2C + D = 1
{A + C - 2D = - 2
{- A + B + D = - 1

{A = - C
{- (- C) + B - 2C + D = 1 → C + B - 2C + D = 1 → - C + B + D = 1
{- C + C - 2D = - 2 → - 2D = - 2 → D = - 2/(- 2) = 1
{- (- C) + B + D = - 1 → C + B + D = - 1

{A = - C = 1
{D = 1
{- C + B + 1 = 1 → B = 1 - 1 + C → B = C = - 1
{C + C + 1 = - 1 → 2C = - 2 → C = - 2/2 = - 1

A = 1, B = - 1, C = - 1, D = 1

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