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