### Video Transcript

Determine the integral of four ๐ฅ squared plus six all squared with respect to ๐ฅ.

The question is asking us to evaluate the integral of the square of a polynomial. And we donโt know how to do this directly. However, we do know how to distribute our square over our parentheses. This will then give us the integral of a polynomial which we can then evaluate.

Thereโs a few different ways of distributing our square over our parentheses. For example, we could use the binomial formula. However, weโre going to do this by using the FOIL method. First, the FOIL method tells us we need to multiply the first two terms together. This gives us four ๐ฅ squared multiplied by four ๐ฅ squared, which we can simplify to give us 16๐ฅ to the fourth power.

Second, the FOIL method tells us weโll want to multiply the outer two terms together. Doing this, we get four ๐ฅ squared multiplied by six, which we can simplify to give us 24๐ฅ squared. Third, the FOIL method tells us we want to multiply the inner two terms together. Which then gives us six times four ๐ฅ squared, which we can calculate is equal to 24๐ฅ squared. Finally, the FOIL method tells us we need to multiply the last two terms together. And this gives us six times six, which is equal to 36.

So, weโve now rewritten our integrand as a polynomial. This means we can evaluate this integral. But, first, we can simplify our integrand. We can add the two terms of 24๐ฅ squared together. Doing this, we get 48๐ฅ squared. So, we now need to evaluate the integral of 16๐ฅ to the fourth power plus 48๐ฅ squared plus 36 with respect to ๐ฅ. This is just the integral of a polynomial. So, weโll do this by splitting the integral of a sum into the sum of the integrals.

Doing this, we get the integral of 16๐ฅ to the fourth power with respect to ๐ฅ plus the integral of 48๐ฅ squared with respect to ๐ฅ plus the integral of 36 with respect to ๐ฅ. Weโre now ready to evaluate each of these integrals by using the power rule for integration.

We recall the power rule for integration tells us if ๐ is not equal to negative one. Then for any constant ๐, the integral of ๐ times ๐ฅ to the ๐th power with respect to ๐ฅ is equal to ๐ times ๐ฅ to the power of ๐ plus one divided by ๐ plus one plus the constant of integration ๐ถ. We add one to our exponent of ๐ฅ and then divide by this new exponent of ๐ฅ.

So, letโs evaluate each of these integrals separately. First, in the integral of 16๐ฅ to the fourth power with respect to ๐ฅ, we can see our exponent of ๐ฅ is four. To integrate this by using the power rule for integration, we want to add one to this exponent of four and then divide by this new exponent. This gives us 16๐ฅ to the power four plus one divided by four plus one. And we then want to add our constant of integration. Weโll call this ๐ถ one. And, of course, we can simplify this since four plus one is equal to five.

We can do something similar to evaluate our second integral. We see the exponent of ๐ฅ is two. And again, we add one to our exponent of two and then divide by this new exponent. This gives us 48 times ๐ฅ to the power of two plus one divided by two plus one. And, of course, we need to add a constant of integration. This time, weโll call this ๐ถ two. And just as we did before, we can simplify this. Two plus one is equal to three. In fact, we can then simplify this even further since 48 divided by three is equal to 16.

So, now, we need to evaluate our third and final integral, the integral of 36 with respect to ๐ฅ. However, thereโs a problem. We donโt seem to be able to use the power rule for integration in this case since thereโs no exponent of ๐ฅ. However, remember that ๐ฅ to the zeroth power is equal to one. So, we can rewrite our integral of 36 as 36 times ๐ฅ to the zeroth power. So, now, we need to evaluate the integral of 36๐ฅ to the zeroth power with respect to ๐ฅ. We can do this by using the power rule for integration. Our exponent of ๐ฅ is equal to zero.

And now, we evaluate this integral using the same method. We add one to our exponent of zero and then divide by this new exponent. This gives us 36๐ฅ to the power of zero plus one divided by zero plus one. And, of course, we need to add a constant of integration. Weโll call this ๐ถ three. And just as we did before, we can simplify this since zero plus one is equal to one. In fact, we can then simplify this further since 36 divided by one is just equal to 36 and ๐ฅ to the first power is just equal to ๐ฅ. This gives us 36๐ฅ.

So, weโve now evaluated all three of our integrals. Thereโs one more thing we can do to simplify this expression. Since ๐ถ one, ๐ถ two, and ๐ถ three are all constants of integration, we can combine all three of these into one new constant of integration which we will call ๐ถ. And this gives us our final answer.

Therefore, we were able to show the integral of four ๐ฅ squared plus six all squared with respect to ๐ฅ is equal to 16๐ฅ to the fifth power divided by five plus 16๐ฅ cubed plus 36๐ฅ plus the constant of integration ๐ถ.