# Solve the following : ∫12x+3x(x+2)⋅dx - Mathematics and Statistics

Sum

Solve the following : int_1^2 (x + 3)/(x (x + 2))*dx

#### Solution

Let I = int_1^2 (x + 3)/(x (x + 2))*dx

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

∴ x + 3 = A(x + 2) + Bx   ...(ii)
Putting x = 0 in (ii), we get
3 = A(0 + 2) + B(0)
∴ 3 = 2A
∴ A = (3)/(2)
Putting x = – 2 in (ii), we get
– 2 + 3 = A(–2 + 2) + B(– 2)
∴ 1 = – 2B
∴ B = (1)/(2)
From (i), we get
(x + 3)/(x(x + 2)) = (3)/(2)*(1)/x - (1)/(2(x + 2)

∴ I = int_1^2[3/(2x) - (1)/(2(x + 2))]*dx

= (3)/(2) int_1^2 (1)/x*dx - (1)/(2) int_1^2 (1)/(x + 2)*dx

= (3)/(2)[log|x|]_1^2 - (1)/(2)[log|x + 2|]_1^2

= (3)/(2)[log |2| - log|1|] - (1)/(2) [log|2 +2| - log|1 + 2|]

= (3)/(2)(log 2 - 0) - (1)/(2)(log4 - log3)

= (3)/(2) log2 - (1)/(2)(log  4/3)

= (1)/(2)(3log2 - log  4/3)

= (1)/(2) log(2^3 xx 3/4)

= (1)/(2)log((8 xx 3)/4)

∴ I = (1)/(2)log6.

Concept: Fundamental Theorem of Integral Calculus
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Chapter 6: Definite Integration - Miscellaneous Exercise 6 [Page 150]

#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 6 Definite Integration
Miscellaneous Exercise 6 | Q 4.02 | Page 150
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