 Good morning friends, I am Purva and today we will work out the following question. Evaluate integral of 1 plus x square upon 1 plus x raised to the power 4 dx. Let us begin with the solution now. Now let us denote integral of 1 plus x square upon 1 plus x raised to the power 4 dx by i. So let i is equal to integral of 1 plus x square upon 1 plus x raised to the power 4 dx. Now this is equal to integral of now dividing the numerator and denominator by x square we get 1 upon x square plus 1 upon 1 upon x square plus x square dx. Now this is equal to integral of 1 upon x square plus 1 upon 1 upon x square plus x square. Now adding and subtracting 2x square into 1 upon x square we get plus 2x square into 1 upon x square minus 2x square into 1 upon x square dx. This is equal to integral of 1 upon x square plus 1 upon now considering 1 upon x square plus x square and minus 2x square into 1 upon x square and applying the formula of a minus b whole square which is equal to a square plus b square minus 2ab we get x minus 1 upon x whole square plus now here we cancel out x square in numerator and denominator and we can write 2 as under root of 2 whole square dx. Now here let x minus 1 upon x is equal to t then differentiating both the sides we get differentiating x we get 1 differentiating minus 1 upon x we get plus 1 upon x square dx is equal to differentiating t we get dt. Now we mark this as 1, we mark this as 2 and we mark this as 3. So putting 2 and 3 in 1 we get i is equal to integral of dt upon t square plus under root of 2 whole square. Now this is equal to 1 upon root 2 into tan inverse t upon root 2 plus c where c is the constant of integration and we get this since we know that integral of dx upon x square plus a square is equal to 1 upon a into tan inverse x upon a plus c. So here taking a is equal to root 2 and x is equal to t we get 1 upon root 2 tan inverse t upon root 2 plus c. Now we know that t is equal to x minus 1 upon x so we get this is equal to 1 upon under root 2 into tan inverse x minus 1 upon x upon root 2 plus c and this is by 2 and we get this is equal to 1 upon under root 2 into tan inverse x square minus 1 upon x into root 2 plus c. So we have got our answer as 1 upon under root 2 into tan inverse x square minus 1 upon x into root 2 plus c. This is our answer. Hope you have understood the solution. Bye and take care.