 Hello and welcome to the session. Let us discuss the following question. It says integrate the following function. The given function is x into under the root of 1 plus 2 x square. Let us now move on to the solution. We have to find the integral of x into under the root of 1 plus 2 x square dx. Now here we see that the derivative of 1 plus 2 x square is 4 x and we want to substitute x dx to use the substitution method. So we put y is equal to 1 plus 2 x square. So dy by dx is equal to 4 x and this implies dy is equal to 4 x dx. Now to substitute x dx for dy we need to divide both sides by 4. So x dx is equal to dy by 4. Now x is dy by 4 and 1 plus 2 x square is y. So substituting all these values the integral becomes under the root y dy by 4. So this is equal to 1 by 4 into integral of y to the power 1 by 2 dy. Now this integral is equal to 1 by 4 into y to the power 1 by 2 plus 1 upon 1 by 2 plus 1 plus 3 as we know that the integral of y to the power n dy is equal to y to the power n plus 1 upon n plus 1 plus c where c is the constant of the integration. Now here n is 1 by 2 so this becomes 1 by 4 into y to the power 3 by 2 upon 3 by 2 plus c which is again equal to 1 by 4 into 2 by 3 into y to the power 3 by 2 plus c. So this is equal to 1 by 6 into y to the power 3 by 2 plus c. Now substitute the value of y here y is 1 plus 2 x square so this becomes 1 by 6 into 1 plus 2 x square to the power 3 by 2 plus c. Hence the integral of the given function is 1 by 6 into 1 plus 2 x square to the power 3 by 2 plus c. And this completes the question. Bye for now. Take care and have a good day.