 Hello and how are you all today? My name is Priyanka and let us discuss this question. It says integrate the following rational functions Now here the function which we need to integrate is 5x upon x plus 1 into x square minus 4 Now let us first simplify the given fraction. So now we have 5x upon x plus 1 into x square minus 4 Simplified and written as 5x upon x plus 1 and x square minus 4 can be written as x plus 2 Into x minus 2, right? So we have the function 5x upon x plus 1 x plus 2 Into x minus 2 is equal to a upon x plus 1 Plus b upon x plus 2 plus c upon x minus 2 taking The calcium We have 5x upon x plus 1 x plus 2 x minus 2 is equal to a into x plus 2 into x minus 2 These are also in bracket Then we have plus b into x plus 1 into x minus 2 plus c into x plus 1 Into x plus 2 The whole divided by x plus 1 into x plus 2 into x minus 2 now further Since the denominators are same so equating the numerators we have 5x is equal to a Into x square minus 4 plus b into x square minus x minus 2 plus c into x square plus 3x plus 2 So now here the coefficient of x square is 0 the coefficient of x is 5 and the coefficient of constants are 0 so we have x square's coefficient from the left right-hand side that is a plus b plus c is equal to 0 then we have for x that is minus b Plus 3c is equal to 5 and then we have the constants that is Minus 4a minus 2b plus 2c equal to 0 So on simplifying these three equations we have the values of a b and C as 5 upon 3 Minus 5 upon 2 and a positive 5 upon 6 therefore the required fraction is 5x upon x plus 1 into x plus 2 into x minus 2 is equal to 5 upon 3 into x plus 1 minus 5 upon 2 into x plus 2 plus 5 upon 6 into x minus 2 now Integrating both the sides we have Integral of now this function can be written as 5x upon x plus 1 in its original form x square minus 4 dx is equal to 5 upon 3 integral of dx upon x plus 1 minus 5 upon 2 integral of dx upon x plus 2 plus 5 upon 6 integral of dx upon x minus 2 This would give the answer as 5 upon 3 log of mod x plus 1 minus 5 upon 2 log of mod x plus 2 plus 5 upon 6 log of mod x minus 2 plus C So here the answer for the question is 5 upon 3 log mod x plus 1 minus 5 upon 2 log mod x plus 2 plus 5 upon 6 log mod x minus 2 Plus C right so this ends the session I hope you have enjoyed the session by and take care