Video: Evaluating Indefinite Integrals

Determine ∫ (2π‘₯ + 1)/(π‘₯Β² + π‘₯ βˆ’ 7) dπ‘₯.

01:09

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. Let’s quickly check this. We can call the denominator 𝑓 of π‘₯. So 𝑓 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 𝑓 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 𝑓 prime of π‘₯ over 𝑓 of π‘₯. And we in fact have a rule for integrating functions of this form. It tells us that the integral of 𝑓 prime of π‘₯ over 𝑓 of π‘₯ with respect to π‘₯ is equal to the natural logarithm of the absolute value of 𝑓 of π‘₯ plus 𝑐. 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 𝑐.

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