Which of the following integrals is improper?

• A. ${\int }_{-1}^1\frac{1}{x+2} dx$
• B. ${\int }_0^1{x}^2 dx$
• C. ${\int }_1^{\infty }\frac{1}{x^2} dx$
• D. ${\int }_0^{2}x+1 dx$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the definition of an improper integral. An integral is considered improper if it involves either an infinite limit of integration or a discontinuity in the integrand within the interval of integration.
2. Step 2: Analyze each integral provided in the problem to determine if it is improper. For example, check if the limits of integration include infinity or if the integrand becomes undefined at any point within the interval.
3. Step 3: For the integral \( \int_{-1}^1 \frac{1}{x+2} \, dx \), observe that the integrand \( \frac{1}{x+2} \) is continuous within the interval \([-1, 1]\), and the limits are finite. Therefore, this integral is not improper.
4. Step 4: For the integral \( \int_0^1 x^2 \, dx \), note that the integrand \( x^2 \) is continuous within the interval \([0, 1]\), and the limits are finite. Thus, this integral is not improper.
5. Step 5: For the integral \( \int_1^\infty \frac{1}{x^2} \, dx \), observe that the upper limit of integration is infinity, which makes this integral improper. Additionally, for \( \int_0^2 (x+1) \, dx \), the integrand \( x+1 \) is continuous within \([0, 2]\), and the limits are finite, so it is not improper.