 Hello and how are you all? The question says integrate the following rational functions and now the function given to us is 2x upon x square plus 1 into x square plus 3. Now if you observe in this question the numerator that is 2x will be a derivative of x square so 2x dx will be equal to dy. So integrating this function can write 2x upon x square plus 1 to x square plus 3 dx can be written now as dy upon y plus 1 into y plus 3. Now here by using the concept of partial fractions we can write the given function as equal to a upon y plus 1 plus b upon y plus 3. Now taking LCM of right inside we have and equating the numerator we have 1 is equal to a into y plus 3 plus b into y plus 1. Now equating the coefficients of y we have a plus b equal to 0 and equating the constants we have 3a plus b equal to 1. Now on solving these two equations we have the values of a and b equal to 1 upon 2 and the value of b is minus 1 upon 2. So writing the given rational function equal to integral minus 1 upon 2 y plus 1 plus 1 upon 2 y plus 3 to dy. Now taking the integration sign separately we have and taking out the constants 1 upon 2 integral dy upon y plus 1 minus 1 upon 2 integral dy upon x plus 3. Now here we can write it as 1 upon 2 log mod 1 y plus 1 minus 1 upon 2 log mod y plus 3 plus c which can also be written as 1 upon 2 log y plus 1 upon y plus 3 plus c. Now on substituting the value of y as x square back we have 1 upon 2 log mod x square plus 1 upon x square plus 3 plus c and this is the required answer to this session. This completes it. Hope you have a good time. Have a nice day.