# Question Video: Evaluating Indefinite Integrals Mathematics • Higher Education

Determine ∫ (2𝑥 + 1)/(𝑥² + 𝑥 − 7) d𝑥.

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### 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.