 Hello and welcome to the session. The given question says evaluate integral x square plus 1 divided by x plus 1 whole square into dx. Let's start with the solution. So here we have to integrate x square plus 1 divided by x plus 1 whole square into dx. Let us denote this integral by i. So this can further be written as integral. x square plus 1 divided by x square plus 2x plus 1 into dx. Now as we can see the degree of the numerator and denominator are both same. So first it is divided. So here we have x square plus 1 and this we have to divide by x square plus 2x plus 1. Multiplying by 1 this gives x square plus 1 plus 2x which further gives this integral as 1 minus 2x divided by x square plus 2x plus 1 into dx. Or this can further be written as integral 1 into dx minus integral 2x divided by x plus 1 whole square into dx. Now let us solve this integral by partial fractions. So we have 2x divided by x plus 1 whole square. This can be written as a divided by x plus 1 plus b divided by x plus 1 whole square. Or this further implies that 2x is equal to a into x plus 1 plus b or we have 2x is equal to ax plus a plus b. On comparing the coefficients we have a is equal to 2 and a plus b is equal to 0 which implies that b is equal to minus of a which gives minus 2. Therefore this can further be written as integral 1 into dx minus integral a divided by x plus 1 and a is 2 so we have 2 divided by x plus 1 into dx plus integral b divided by x plus 1 whole square and b here is minus 2 so we have minus 2 divided by x plus 1 whole square into dx. Now this is further equal to x minus 2 times log mod x plus 1 plus 2 times of integral x plus 1 raised to the power minus 2 into dx which further implies that x minus 2 log mod x plus 1 plus 2 times of x plus 1 raised to the power minus 1 divided by minus 1 plus a constant c or is this further equal to x minus 2 times of log mod x plus 1 minus 2 times of 1 divided by x plus 1 plus a constant therefore the value of the given integral is x minus 2 log mod x plus 1 minus 2 times of 1 divided by x plus 1 plus a constant so this completes the session bye and take care