Math, asked by jaspreet79862jaspree, 4 months ago

evaluate x /x+1 x+2 x+3 dx

Answers

Answered by amansharma264

EXPLANATION.

⇒ ∫X/(x + 1)(x + 2)(x + 3).dx.

As we know that,

Partial fraction method is apply only when coefficient of Denominator > coefficient of Numerator.

⇒ ∫x/(x + 1)(x + 2)(x + 3).dx = A/(x + 1) + B/(x + 2) + C/(x + 3).

Taking L.C.M on both sides, we get.

⇒ x/(x + 1)(x + 2)(x + 3) = A(x + 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x + 2).

⇒ x = A(x + 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x + 2).

Put the value of x = - 3 in equation, we get.

⇒ (-3) = A(- 3 + 2)(- 3 + 3) + B(- 3 + 1)(- 3 + 3) + C(- 3 + 1)(- 3 + 2).

⇒ (-3) = 0 + 0 + C(- 2)(- 1).

⇒ (-3) = C(2).

⇒ C = -3/2.

Put the value of x = - 2 in equation, we get.

⇒ (-2) = A(- 2 + 2)(- 2 + 3) + B(- 2 + 1)(- 2 + 3) + C(- 2 + 1)(- 2 + 2).

⇒ (-2) = 0 + B(- 1)(1) + 0.

⇒ (-2) = (-B).

⇒ B = 2.

Put the value of x = - 1 in equation, we get.

⇒ (-1) = A(- 1 + 2)(- 1 + 3) + B(- 1 + 1)(- 1 + 3) + C(- 1 + 1)(- 1 + 2).

⇒ (-1) = A(1)(2) + 0 + 0.

⇒ (-1) = 2A.

⇒ A = -1/2.

Put the value in this equation, we get.

⇒ ∫x/(x + 1)(x + 2)(x + 3).dx = ∫A/(x + 1).dx + ∫B/(x + 2).dx + ∫C/(x + 3).dx.

⇒ ∫A/(x + 1).dx + ∫B/(x + 2).dx + ∫C/(x + 3).dx.

⇒ ∫-1/2/(x + 1).dx + ∫2/(x + 2).dx + ∫-3/2/(x + 3).dx.

⇒ ∫-1.dx/2(x + 1) + ∫2/(x + 2).dx + ∫-3/2(x + 3).dx.

⇒ -1/2∫1/(x + 1).dx + 2∫1/(x + 2) + (-3/2)∫1/(x + 3).dx.

⇒ -1/2㏑(x + 1) + 2㏑(x + 2) -3/2㏑(x + 3) + c.

MORE INFORMATION.

$\sf \implies \int sin^{m} x cos^{n} x dx$

(a) = If m and n are even integer then convert in terms of multiple angle.

(b) = m is odd and n is even then put cos(x) = t.

(d) = if, m and n both odd then put either sin(x) = t or cos(x) = t.

(e) = if m + n = -ve and even, then rearrange Q, in terms of ''tan(x)'' and put t = tan(x).

Answered by Rosy44444

Answer: