Solve this integration::
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Answers
EXPLANATION.
⇒ ∫(xdx)/(x - 1)²(x + 2).
As we know that,
If coefficient of Numerator < Denominator then,
Apply partial fractions method in this equation, we get.
⇒ ∫(xdx)/(x - 1)²(x + 2). = A/(x - 1) + B/(x - 1)² + C/(x + 2).
⇒ x = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)².
Put the value of x = 1 in the equation, we get.
⇒ 1 = A(1 - 1)(1 + 2) + B(1 + 2) + C(1 - 1)².
⇒ 1 = 0 + B(3) + 0.
⇒ 1 = 3B.
⇒ B = 1/3.
Put the value of x = - 2 in the equation, we get.
⇒ - 2 = A(- 2 - 1)(- 2 + 2) + B(- 2 + 2) + C(- 2 - 1)².
⇒ - 2 = 0 + 0 + C(- 3)².
⇒ - 2 = 9C.
⇒ C = - 2/9.
Put the value of x = 0 in the equation, we get.
⇒ 0 = A(0 - 1)(0 + 2) + B(0 + 2) + C(0 - 1)².
⇒ 0 = A(- 1)(2) + B(2) + C(- 1)².
⇒ 0 = -2A + 2B + C.
Put the value of b = 1/3 and c = - 2/9 in the equation, we get.
⇒ 0 = - 2A + 2(1/3) + (-2/9).
⇒ 0 = - 2A + (2/3) - (2/9).
⇒ 0 = [(- 18A) + 6 - 2]/(9).
⇒ 0 = - 18A + 6 - 2.
⇒ 0 = - 18A + 4.
⇒ 18A = 4.
⇒ 9A = 2.
⇒ A = 2/9.
Values of A = 2/9, B = 1/3 and C = -2/9.
Put the values in the equation, we get.
⇒ ∫(xdx)/(x - 1)²(x + 2). = ∫A/(x - 1)dx + ∫B/(x - 1)²dx + ∫C/(x + 2)dx.
⇒ ∫2/9(x - 1) dx + ∫dx/3(x - 1)² + ∫-2/9(x + 2) dx.
⇒ 2/9㏑|(x - 1)| + 1/3㏑|(x - 1)² - 2/9㏑|(x + 2)| + C.
⇒ ∫(xdx)/(x - 1)²(x + 2) = 2/9㏑|(x - 1)| + 1/3㏑|(x - 1)² - 2/9㏑|(x + 2)| + C.
MORE INFORMATION.
Integration by parts.
(1) If u and v are two functions of x then,
∫(u.v)dx = u(∫vdx) - ∫[(du/dx).(∫vdx)]dx.
From the first letter of the word.
I = Inverse trigonometric functions.
L = Logarithmic functions.
A = Algebraic functions.
T = Trigonometric functions.
E = Exponential functions.
We get a word = ILATE.
First arrange the functions in the order according to letters of this word and then integrate by parts.
(2) If the integral is of the form,
∫eˣ[f(x) + f'(x)]dx = eˣf(x) + c.
(3) If the integral is of the form,
∫[xf'(x) + f(x)]dx = x f(x) + c.