### Video Transcript

Determine the integral of two ๐ฅ divided by seven ๐ฅ squared plus one with respect to ๐ฅ.

Weโre asked to evaluate the integral of the quotient of two polynomials. Thatโs a rational function. And thereโs a few different ways we know how to try and integrate rational functions. For example, we might want to try using partial fractions. However, in this case, our denominator doesnโt factor, so this wonโt work. We could also try something like substitution. However, in actual fact, thereโs a simpler method.

Letโs call the function in our denominator ๐ of ๐ฅ. So thatโs seven ๐ฅ squared plus one. We want to check what ๐ prime of ๐ฅ is. We can do this by using the power rule for differentiation. We get that ๐ prime of ๐ฅ is 14๐ฅ. And we can see that 14๐ฅ is a linear multiple of our numerator. So we can integrate this by using one of our integral rules. We need to recall the following. The integral of ๐ prime of ๐ฅ over ๐ of ๐ฅ with respect to ๐ฅ is equal to the natural logarithm of the absolute value of ๐ of ๐ฅ plus the constant of integration ๐ถ.

Now in this case, our expression is not quite in this form. If we set ๐ of ๐ฅ to be our denominator, weโve shown the derivative of seven ๐ฅ squared plus one with respect to ๐ฅ is 14๐ฅ. However, our numerator is two ๐ฅ. We need our numerator to be 14๐ฅ, so what we could do is multiply this by seven and then divide our entire integral by seven. Multiplying by seven and dividing by seven is the same as multiplying by one, so it wonโt change our value. However, now we can see our numerator will be 14๐ฅ. Thatโs ๐ prime of ๐ฅ.

So now we can evaluate this integral by using our integral rule. Weโll set ๐ of ๐ฅ to be seven ๐ฅ squared plus one. This gives us one-seventh times the natural logarithm of the absolute value of seven ๐ฅ squared plus one plus a constant of integration weโll call ๐ด. And we can simplify this expression. First, weโll distribute one-seventh over our parentheses. This gives us one-seventh times the natural logarithm of the absolute value of seven ๐ฅ squared plus one plus ๐ด over seven. And remember, ๐ด is just a constant, so ๐ด over seven is also a constant. We can just call this a new constant called ๐ถ.

And by doing this, we get our final answer. Therefore, we were able to show the integral of two ๐ฅ over seven ๐ฅ squared plus one with respect to ๐ฅ is equal to one-seventh times the natural logarithm of the absolute value of seven ๐ฅ squared plus one plus our constant of integration ๐ถ.