# Video: Finding the Integration of a Polynomial

Determine ∫ (𝑥 − 6)(𝑥 − 5)(𝑥 − 3) d𝑥.

02:14

### Video Transcript

Determine the indefinite integral of 𝑥 minus six multiplied by 𝑥 minus five multiplied by 𝑥 minus three.

Let’s start by expanding the brackets. Expanding the first two sets of brackets, we get 𝑥 squared minus 11𝑥 plus 30. And we then multiply this by 𝑥 minus three. We obtain the indefinite integral of 𝑥 cubed minus 14𝑥 squared plus 63𝑥 minus 90 which is in fact a polynomial. And we can use the power rule for integration to integrate this term by term. The power rule tells us that the integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to 𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus 𝐶. If instead we were integrating 𝑥 to the power of 𝑛 multiplied by some constant 𝑎, then since we can factor a constant out of our integral, then this would simply be equal to 𝑎 multiplied by 𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus 𝐶.

Now, you may be wondering why we haven’t multiplied the 𝐶 by 𝑎. And that’s because 𝑎 is also a constant. Therefore, 𝑎 multiplied by 𝐶 is a constant. And we can just rename this new constant to be 𝐶. So now, let’s apply this rule to our integral term by term. The first term is 𝑥 cubed. Therefore, 𝑛 is equal to three. We increase the power by one and divide by the new power to get 𝑥 to the power of four over four. The next term is negative 14𝑥 squared. Negative 14 is just a constant. So that will remain. Our power is two. So 𝑛 is two. We increase the power by one to get 𝑥 cubed and divide by the new power.

On next term, we have 63𝑥. So we can start by writing in our constant of 63. We then note that 𝑥 is equal to 𝑥 to the power of one. So 𝑛 is equal to one. So when we integrate this, we get 𝑥 squared over two. For our final term, we have negative 90. And we know that this can also be written as negative 90 multiplied by 𝑥 to the power of zero since 𝑥 to the power of zero is just one. So now, we can integrate it. We start by writing our constant of negative 90. Since our power of 𝑥 is zero, we increase the power by one giving us 𝑥 to the power of one and divide by the new power. So that’s dividing by one. And we miss then adding our constant of integration 𝐶.

We can write this out a little neater for our solution which is 𝑥 to the power of four over four minus 14𝑥 cubed over three plus 63𝑥 squared over two minus 90𝑥 plus 𝐶.