### Video Transcript

Determine the integral of three root 𝑥 plus seven over nine root 𝑥 all squared with respect to 𝑥.

We’re asked to evaluate the integral of a square. And we see we can’t evaluate this integral directly. However, we do know how to distribute the square over our parentheses. We can do this by using binomial expansion or the FOIL method. And in doing this, we see we will get terms which are radicals, and we know how to integrate these.

So the first thing we need to do is simplify our integrand by distributing the square over our parentheses. We’ll do this by using binomial expansion. To get our first term, we just need to square the first term inside of our binomial. This gives us three root 𝑥 all squared. And we can evaluate this by using our laws of exponents. It’s equal to nine 𝑥.

The next term in our expansion will be two multiplied by the product of both of the terms in our binomial. This gives us two times three root 𝑥 multiplied by seven over nine root 𝑥. And we can simplify. Root 𝑥 divided by root 𝑥 gives us one. And we can simplify this further. We can cancel the shared factor of three in our numerator and our denominator and simplify two multiplied by seven to give us 14. So our second term simplifies to give us 14 over three.

The last thing we need to do is square the final term in our binomial. This gives us seven over nine root 𝑥 all squared, which we can simplify by using our laws of exponents to give us 49 over 81𝑥. So by using binomial expansion, we were able to rewrite our integral as the integral of nine 𝑥 plus 14 over three plus 49 over 81𝑥 with respect to 𝑥. And we see we can evaluate the integral of each of these terms separately.

We’ll integrate our first term by using the power rule for integration. We add one to our exponent of 𝑥 and then divide by this new exponent. This gives us nine 𝑥 squared over two. We can see that our second term is a constant. One way of evaluating the integral of a constant is to use the power rule for integration. We would get 14𝑥 divided by three.

Finally, in our third term, we can see we’re dividing by 𝑥. We know by using our laws of exponents this is the same as having a factor of 𝑥 to the negative one in our numerator. And because the exponent of 𝑥 is negative one, this means we can’t use the power rule. Instead, we’re going to have to use our rules for integrating reciprocal functions. We recall, for any real constant 𝑎, the integral of 𝑎 times 𝑥 to the power of negative one with respect to 𝑥 is equal to 𝑎 times the natural logarithm of the absolute value of 𝑥 plus a constant of integration 𝐶. In our case, the value of 𝑎 is 49 over 81. So integrating our third term with respect to 𝑥, we get 49 over 81 times the natural logarithm of the absolute value of 𝑥 plus 𝐶. And this is our final answer.

Therefore, by using binomial expansion, the power rule, and our rules for integrating reciprocal functions, we were able to show the integral of three root 𝑥 plus seven over nine root 𝑥 all squared with respect to 𝑥 is equal to nine 𝑥 squared over two plus 14𝑥 over three plus 49 over 81 times the natural logarithm of the absolute value of 𝑥 plus 𝐶.