 Hi and welcome to the session. I am Purva and I will help you with the following question. Integrate the function 1 upon x square into x raised to the power 4 plus 1 this whole raised to the power 3 by 4. Now we begin with the solution. So we denote this function by i. So we have i is equal to integral 1 upon x square into x raised to the power 4 plus 1 this whole raised to the power 3 by 4 dx. We can write this as this is equal to integral 1 upon x square. Now taking x raised to the power 4 common from here we get x raised to the power 4 to the whole power 3 by 4 into. Now taking x raised to the power 4 common here we get 1. So 1 plus and here we get 1 upon x raised to the power 4. So 1 upon x raised to the power 4 to the whole power 3 by 4 dx. Now cancelling here 4 and 4 we are left with x cube. This is equal to integral 1 upon now we know that when bases are same then powers are added. So we can write x square into x cube as x raised to the power 5. So we have x raised to the power 5 into 1 plus 1 upon x raised to the power 4 whole power 3 by 4 dx. Now put 1 plus 1 upon x raised to the power 4 equal to t. So we have put 1 plus 1 upon x raised to the power 4 equal to t. So we can write this as this implies 1 plus taking this numerator we get x raised to the power minus 4 is equal to t. Differentiating we get differentiating we get this 1 will go. So we will have minus 4 into x raised to the power minus 4 minus 1 dx is equal to dt. This implies minus 4 into x raised to the power minus 5 because minus 4 minus 1 is minus 5 dx is equal to dt and this further implies minus 4 upon x raised to the power 5 dx is equal to dt or we can write this as dx upon x raised to the power 5 is equal to dt upon minus 4. Right? Putting all these values in i we get i is equal to minus 1 by 4 integral dt upon t raised to the power 3 by 4 or we can write this as this is equal to minus 1 by 4 integral t raised to the power minus 3 by 4 dt. Integrating we get i is equal to minus 1 by 4 t raised to the power minus 3 by 4 plus 1 upon minus 3 by 4 plus 1 this is into plus c. Now we have got this because we know that since x raised to the power n is equal to x raised to the power n plus 1 divided by n plus 1 right? So we have got this. This is equal to minus 1 by 4 into t raised to the power 1 by 4 because minus 3 by 4 plus 1 is 1 by 4 upon 1 by 4 plus c and this is further equal to now cancelling this 4 and this 4 we get minus t raised to the power 1 by 4 plus c. This is equal to minus now we have put t is equal to 1 plus 1 by x raised to the power 4 so we get 1 plus 1 upon x raised to the power 4 this whole to the power 1 by 4 plus c since t is equal to 1 plus 1 upon x raised to the power 4. So we get our answer as minus 1 plus 1 upon x raised to the power 4 whole raised to the power 1 by 4 plus c. So this is the answer of the question. Hope you have understood the solution. Take care and God bless you.