 Hello friends, let's discuss the following question, it says integrate the following function. The given function is 4x plus 2 into under the root of x square plus x plus 1. Let's now move on to the solution. We have to integrate 4x plus 2 into under the root of x square plus x plus 1 dx. Now this integral can be written as taking to common we have 2x plus 1 into under the root of x square plus x plus 1 dx. Which is again equal to, now we see that the integral of x square plus x plus 1 is 2x plus 1. So we put y is equal to x square plus x plus 1. So dy by dx is equal to 2x plus 1 and this implies dy is equal to 2x plus 1 dx. So 2x plus 1 into dx is dy and y is equal to x square plus x plus 1. So we substitute all these value in the integral so the integral becomes 2 into under the root y. Because y is x square plus x plus 1 and 2x plus 1 into dx is dy. So this is equal to 2 into integral y to the power 1 by 2 dy. And this is equal to 2 into y to the power 1 by 2 plus 1 upon 1 by 2 plus 1 plus c. As we know that the integral of y to the power n dy is y to the power n plus 1 upon n plus 1 plus c. Now here n is 1 by 2. Again this is equal to y to the power 3 by 2 upon 3 by 2 plus c. And this is again equal to 4 by 3 into y to the power 3 by 2 plus c. Now substituting the value of y, y is x square plus x plus 1 so it becomes 4 by 3 into x square plus x plus 1 to the power 3 by 2 plus c. Hence the integral of the given function is 4 by 3 into x square plus x plus 1 whole to the power 3 by 2 plus c. And this completes the question. Bye for now. Take care. Have a good day.