# Solve the following : ∫23x(x+2)(x+3)⋅dx - Mathematics and Statistics

Sum

Solve the following : int_2^3 x/((x + 2)(x + 3))*dx

#### Solution

Let I = int_2^3 x/((x + 2)(x + 3))*dx

Let x/((x + 2)x + 3) = "A"/(x + 2) + "B"/(x + 3)  ...(i)

∴ x = A(x + 3) + B(x + 2)        ...(ii)
Putting x = – 3 in (ii) we get
– 2 = A
∴ B = 3
Putting x = – 2 in (ii),we get
– 2 = A
∴ A = – 2
From (i), we get
x/((x + 2(x + 3))) = (-2)/(x + 2) + (3)/(x + 3)

∴ I = int_2^3 [(-2)/(x + 2) + 3/(x + 3)]*dx

= -2int_2^3 (1)/(x + 2)*dx + 3 int_2^3 (1)/(x + 3)*dx

= -2[log|x + 2|]_2^3 + 3[log|x + 3|]_2^3

= -2log[log 5 – log 4] + 3[log 6 – log 5]

= -2[log(5/4)] + 3[log(6/5)]

= 3log(6/5) - 2log(5/4)

= log(6/5)^2 - 2log(5/4)^2

= log(216/125) - log(25/16)

= log(216/125 xx 16/25)

∴ I = log(3456/3125).

Concept: Fundamental Theorem of Integral Calculus
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 6 Definite Integration
Miscellaneous Exercise 6 | Q 4.01 | Page 150