 Hello friends and how are you all today? My name is Priyanka and I shall be helping you with the following question says integrate The following rational functions now here the function which you need to integrate is 2x upon x square plus 3x plus 2 so We can write this function as 2x upon x plus 1 Into x plus 2 Right now further we have 2x upon x plus 1 Into x plus 2 written as a upon x plus 1 plus b upon x plus 2 now on taking the LCM we have the same denominator and Now in the numerator a will get multiplied with x plus 2 whereas b will get multiplied with x plus 1 now on comparing the coefficients We have the coefficients of x as a plus b is equal to 2 and the constants that is 2a plus b equal to 0 Now on solving these two equation, we have the value of a and b equal to the value of b as 4 and a as minus 2 So now We can write The given rational function as to minus 2 upon x plus 1 plus 4 upon x plus 2 now integrating both the sides we have integral of 2x dx upon x plus 1 into x plus 2 Equal to now taking the integral sign separately we have minus 2 Integral dx upon x plus 1 plus 4 integral dx upon x plus 2 Which can be written as minus 2? log of mod x plus 1 plus 4 log x plus 2 Plus c so the required answer for the section is equal to 4 log Mod x plus 2 minus 2 log mod x plus 1 Plus c right this ends the session. Hope you understood it well and have a nice day