Video Transcript
Determine the indefinite integral
of two 𝑥 plus one over 𝑥 squared plus 𝑥 minus seven with respect to 𝑥.
When we look at the integrand of
this integral, we can notice that the numerator looks a lot like the derivative of
the denominator. Let’s quickly check this. We can call the denominator 𝑓 of
𝑥. So 𝑓 of 𝑥 is equal to 𝑥 squared
plus 𝑥 minus seven. And now we can differentiate
this. Using the power rule on each term,
we obtain that 𝑓 prime of 𝑥 is equal to two 𝑥 plus one. Which is equal to the numerator of
our fraction. Therefore, our integrand is of the
form 𝑓 prime of 𝑥 over 𝑓 of 𝑥. And we in fact have a rule for
integrating functions of this form. It tells us that the integral of 𝑓
prime of 𝑥 over 𝑓 of 𝑥 with respect to 𝑥 is equal to the natural logarithm of
the absolute value of 𝑓 of 𝑥 plus 𝑐. Using this rule, we obtain that the
integral of two 𝑥 plus one over 𝑥 squared plus 𝑥 minus seven with respect to 𝑥
is equal to the natural logarithm of the absolute value of 𝑥 squared plus 𝑥 minus
seven plus our constant of integration 𝑐.
