# How to use integration by parts to find the definite integral for the problems 1. (3x^2 +5x+6)/(x^3 -16x) 2. (x+7)/(x^2 +7x +10)

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1) I = (3x^2 +5x+6)/(x^3 -16x) = (3x^2 +5x+6)/{x*(x^2 -16)}

= (3x^2 +5x+6)/(x^3 -16x) = (3x^2 +5x+6)/{x*(x-4)*(x+4)}

Let (3x^2 +5x+6)/(x^3 -16x) = {A/x} + {B/(x-4)} + {C/(x+4)}

Thus, (3x^2 +5x+6)/(x^3 -16x) = [A*{(x+4)*(x-4)} + B*{x*(x+4)} + C*{(x-4)*x}]/{(x^3 -16x)}

or, (3x^2 +5x+6) = [A*{(x+4)*(x-4)} + B*{x*(x+4)} + C*{(x-4)*x}]

On solving we get,

A = -3/8 ; B = 37/16 & C = 17/16

Thus, I = -{(3/8)*(1/x)} + {(37/16)*(1/(x-4))} + {(17/16)*(1/(x+4))}

Integrating I we get:- -(3/8)lnx + (37/16)ln(x-4) + (17/16)*ln(x+4) + c

where 'c' = constant of integration

2) I = (x+7)/(x^2 +7x +10) = (x+7)/{(x+2)*(x+5)}

Let (x+7)/{(x+2)*(x+5)} = {A/(x+2)} + {B/(x+5)}

Thus, (x+7)/{(x+2)*(x+5)} = [{A*(x+5)} + {B*(x+2)}]/{(x+2)*(x+5)}

or, x+7 = [{A*(x+5)} + {B*(x+2)}]

On solving we get,

A = 5/3 & B = -2/3

Thus, I = {(5/3)*(1/(x+2))} - {(2/3)*(1/(x+5))}

On integrating I we get

(5/3)*ln(x+2) - (2/3)*ln(x+5)