Question
12 Multiple Choice 1 point Do not integrate, but state which method(s) can be used to evaluate the integral: \( \int(x+1) /\left(x^{2}+2 x-8\right) d x \)
Ask by Ruiz Mathis. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral can be evaluated using either u-substitution or partial fraction decomposition.
Solution
We start by factoring the denominator:
x² + 2x – 8 = (x – 2)(x + 4).
Now, notice that the derivative of x² + 2x – 8 is 2x + 2, which equals 2(x + 1). Since the numerator is x + 1, you can rewrite the integrand as
½ · [2(x + 1)]/(x² + 2x – 8).
This observation shows that if you let u = x² + 2x – 8, then du = 2(x + 1) dx, making the substitution method perfectly applicable.
Alternatively, since the integrand, after factoring the denominator, becomes a rational function with linear factors in the denominator, partial fractions can be used.
Thus, the methods that can be used to evaluate the integral are either u-substitution or partial fraction decomposition.
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Simplify this solution The Deep Dive
To evaluate the integral \( \int \frac{x+1}{x^2 + 2x - 8} \, dx \), you can use the method of partial fraction decomposition. This involves expressing the integrand as a sum of simpler fractions, making integration more straightforward. Another helpful method is substitution, particularly if you can identify a u-substitution that simplifies the function inside the integral. In this case, factoring the denominator first could provide a clearer path for choosing an appropriate substitution or applying partial fractions.
