 Hello friends and how are you all today? My name is Priyanka and the question says integrant the following rational functions. Now here the function which is given to us is 2 upon 1 minus x into 1 plus x whole, 1 plus x square. Now since the integrant is a proper rational function, so decomposing the rational function into partial fractions we get a upon 1 minus x plus bx plus c upon 1 plus x square since it is a quadratic. So on taking the LCM and equating the numerator we have 2 is equal to a 1 plus x square plus bx plus c into 1 minus x let this be the first equation. Now putting value of x equal to 1 in the first equation we get 2 is equal to a 1 plus 1 plus b into 1 plus c 1 minus 1 it will be equal to 0. So we have the value of 2 equal to 2a plus 0 that is the value of a is equal to 1. Now equating the coefficient of x cube and the constant terms on both the sides of equation 1 we get a minus b is equal to 0 that further implies the value of a is equal to value of b the value of a is obtained above as 1 so value of b will also be equal to 1. Also we have equating the constants a plus c is equal to 2 so we have 2 plus c is equal to sorry 1 plus c is equal to 2 that further implies the value of c is equal to 1. Therefore the fraction will become 2 upon 1 minus x into 1 plus x square is equal to 1 upon 1 minus x that was a upon 1 minus x plus bx so the value of b is 1 so we will have x plus c that is 1 upon 1 plus x square. Now the required integrand is 2 upon 1 minus x into 1 plus x square dx is equal to integral of 1 upon 1 minus x plus x plus 1 upon 1 plus x square to dx. Now the required answer will be like this it will be equal to 1 upon 1 minus x dx plus x dx upon 1 plus x square plus dx upon 1 plus x square that is further equal to minus log mod 1 minus x plus 1 upon 2 log 1 plus x square plus the integral of 1 plus x square is tan inverse x plus c so the required answer to the section is minus log mod 1 minus x plus 1 by 2 log mod 1 plus x square plus tan inverse c plus c sorry tan inverse x plus c right so this completes the session hope you understood the whole concept well and have a very nice day.