# Question Video: Evaluating Indefinite Integrals Mathematics • Higher Education

Determine β« (2π₯ + 1)/(π₯Β² + π₯ β 7) dπ₯.

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### 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.
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