 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says find the following integral that is integral of root x into 3x squared plus 2x plus 3 dx. So let us start with the solution to the question. We have to find integral of root x into 3x squared plus 2x plus 3 dx. Now on opening the brackets we get integral of 3x square into root x dx plus integral of x into root x dx plus integral of, we know that root x is same as x raised to power 1 by 2 and we see that from these three integrals we have three constant that is 3, 2 and 3. So they come out of the integral as it is. We will have 3 into integral of x square into x raised to power 1 by 2 dx plus 2 into integral of x into root x or x raised to power 1 by 2 dx plus 3 into integral of x raised to power 1 by 2 dx. This can be written as 3 into integral of x raised to power 2 plus 1 by 2 is 5 by 2 so 5 by 2 dx plus 2 into integral of x raised to power 1 plus 1 by 2 is 3 by 2 dx plus 3 into integral of x raised to power 1 by 2 dx. Now this is equal to 3 into x raised to power 5 by 2 plus 1 divided by 5 by 2 plus 1 plus 2 into x raised to power 3 by 2 plus 1 divided by 3 by 2 plus 1 plus 3 into x raised to power 1 by 2 plus 1 divided by 1 by 2 plus 1 plus a constant c. Now this happens because we see that integral of x raised to power n dx is equal to x raised to power n plus 1 divided by n plus 1 plus a constant c where n is not equal to minus 1. So in the first case n was 5 by 2 in second integral n was 3 by 2 and in third integral n was 1 by 2. So on applying this formula we get this. Let us simplify this. We get 3 into x raised to power 5 by 2 plus 1 is 7 by 2 divided by 7 by 2 plus 2 into x raised to power 3 by 2 plus 1 is 5 by 2 divided by 5 by 2 plus 3 into x raised to power 1 plus 1 by 2 is 3 by 2 divided by 3 by 2 plus a constant c. This can be further written as we take this to the numerator so we get 6 into x raised to power 7 by 2 divided by 7 plus 4 by 5 into x raised to power 5 by 2 plus 2 goes to the numerator we have 2 into x raised to power 3 by 2 plus a constant c. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.