00:01
In this problem we are given double integral r x over 2 plus x by the whole square da.
00:16
We are given the region r as x comma y for x less than or equals to 4 and for y 1 less than or equals to y less than or equals to 2.
00:33
So first let us write this integral i is equals to double integral so x ranges from 0 to 4 and y ranges from 1 to 2 x over 2 plus x by the whole square so the region dy dx.
00:56
So now next we have to use u substitution where u is equals to x y over 2 sorry x y plus 2 so du is equals to x dy therefore du over dy is equals to x.
01:21
So now substituting and changing this we get integral 0 to 4 integral 1 to 2 du over u square dx.
01:34
So now first let us integrate with respect to u so we get integral 0 to 4 negative 1 over u 1 to 2 dx.
01:49
So this becomes 0 to 4 negative 1 over 2 sorry we have to replace this u by x y plus 2 so x y plus 2 1 to 2 dx.
02:09
So now we will get integral 0 to 4 negative 1 over so now we have to replace y by 2 so we have 2x plus 2 negative of negative 1 over x plus 2 dx.
02:31
So now solving this we get integral 0 to 4 x over 2x squared plus 6x plus 4 dx.
02:49
So we have solved only one integral till now we have to move on to dx integration...