 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says find the following integral that is integral of x square plus 3x plus 4 divided by root x dx. So let us find the solution to this question. We see that it is given to us integral of x square plus 3x plus 4 divided by root x dx. Now this can be written as integral of x square by root x dx plus integral of 3x by root x dx plus integral of 4 by root x dx. Now what we have done here is we have taken the root x in the denominator with each numerator and then we have separated the terms. Now this can be further written as integral of x square by x raise to power 1 by 2 dx plus integral of 3x divided by x raise to power 1 by 2 dx plus integral of 4 divided by x raise to power 1 by 2 dx. Now taking x raise to power 1 by 2 in the numerator in each term we get integral of x square into x raise to power minus 1 by 2 dx plus 3 into integral of x into x raise to power minus 1 by 2 dx. Now since 3 is a constant so it comes out of the integral sign as it is plus 4 into integral of x raise to power minus 1 by 2 dx. Now we know that whenever bases are same then in multiplication the powers get added so we get integral of x raise to power 2 minus 1 by 2 that is 3 by 2 so x raise to power 3 by 2 dx plus 3 into integral of x raise to power 1 minus 1 by 2 is 1 by 2 dx plus 4 into integral of x raise to power minus 1 by 2 dx. Now this will be equal to x raise to power 3 by 2 plus 1 divided by 3 by 2 plus 1 plus x raise to power 1 by 2 plus 1 divided by 1 by 2 plus 1 plus 4 into x raise to power minus 1 by 2 plus 1 divided by minus 1 by 2 plus 1 plus a constant c. Now this we get because we know that integral of x raise to power n dx is given by x raise 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 term n was 3 by 2 in the second n was 1 by 2 and in the third n was minus 1 by 2. So on further simplification we get x raise to power now 3 by 2 plus 1 is 5 by 2 so x raise to power 5 by 2 by 5 by 2 plus 3 into x raise to power 1 plus 1 by 2 is 3 by 2 divided by 3 by 2 plus 4 into x raise to power 1 by 2 divided by 1 by 2 plus a constant c. On further simplification we can write it as 5 by 2, 3 by 2 and 1 by 2 we take them to the numerator and they become 2 by 5 into x raise to power 5 by 2 plus 2 into x raise to power 3 by 2 because this 3 gets cancelled with this 3 plus 8 into x raise to power 1 by 2 plus a constant c. This can be further written as root x divided by 5 into square plus 10x plus 40 plus a constant c. Now what we have done here is we have taken x raise to power 1 by 2 common also 1 by 5 common from each term so we get this. So our answer to this question is square root of x divided by 5 into 2x square plus 10x plus 40 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.